Products of compact lines Gonzalo Mart nez Cervantes University of - PowerPoint PPT Presentation

How do intervals and rectangles differ?

How do intervals and rectangles differ?

How do intervals and rectangles differ?

How do intervals and rectangles differ?

How do intervals and rectangles differ? The blue cover has 3 intervals.

How do intervals and rectangles differ? The blue cover has 3 intervals. The red cover also has 3 intervals.

How do intervals and rectangles differ? The blue cover has 3 intervals. The red cover also has 3 intervals. We can find a finer cover which has 5 intervals.

How do intervals and rectangles differ? The blue cover has 3 intervals. The red cover also has 3 intervals. We can In general, if we have find a finer cover which has 5 intervals. C 1 , . . . , C k different covers of a compact line (made with intervals) then we can always find a finer cover C such that |C| ≤ 2( |C 1 | + . . . + |C k | ) .

How do intervals and rectangles differ? The blue cover has 3 intervals. The red cover also has 3 intervals. We can In general, if we have find a finer cover which has 5 intervals. C 1 , . . . , C k different covers of a compact line (made with intervals) then we can always find a finer cover C such that |C| ≤ 2( |C 1 | + . . . + |C k | ) .

How do intervals and rectangles differ? The blue cover has 3 intervals. The red cover also has 3 intervals. We can In general, if we have find a finer cover which has 5 intervals. C 1 , . . . , C k different covers of a compact line (made with intervals) then we can always find a finer cover C such that |C| ≤ 2( |C 1 | + . . . + |C k | ) .

How do intervals and rectangles differ? The blue cover has 3 intervals. The red cover also has 3 intervals. We can In general, if we have find a finer cover which has 5 intervals. C 1 , . . . , C k different covers of a compact line (made with intervals) then we can always find a finer cover C such that |C| ≤ 2( |C 1 | + . . . + |C k | ) .

How do intervals and rectangles differ? The blue cover has 3 intervals. The red cover also has 3 intervals. We can In general, if we have find a finer cover which has 5 intervals. C 1 , . . . , C k different covers of a compact line (made with intervals) then we can always find a finer cover C such that |C| ≤ 2( |C 1 | + . . . + |C k | ) .

How do intervals and rectangles differ? The blue cover has 3 intervals. The red cover also has 3 intervals. We can In general, if we have find a finer cover which has 5 intervals. C 1 , . . . , C k different covers of a compact line (made with intervals) then we can always find a finer cover C such that |C| ≤ 2( |C 1 | + . . . + |C k | ) .

How do intervals and rectangles differ? The blue cover has 3 intervals. The red cover also has 3 intervals. We can In general, if we have find a finer cover which has 5 intervals. C 1 , . . . , C k different covers of a compact line (made with intervals) then we can always find a finer cover C such that |C| ≤ 2( |C 1 | + . . . + |C k | ) . The blue cover has 3 rectangles.

How do intervals and rectangles differ? The blue cover has 3 intervals. The red cover also has 3 intervals. We can In general, if we have find a finer cover which has 5 intervals. C 1 , . . . , C k different covers of a compact line (made with intervals) then we can always find a finer cover C such that |C| ≤ 2( |C 1 | + . . . + |C k | ) . The blue cover has 3 rectangles. The red cover also has 3 rectangles.

How do intervals and rectangles differ? The blue cover has 3 intervals. The red cover also has 3 intervals. We can In general, if we have find a finer cover which has 5 intervals. C 1 , . . . , C k different covers of a compact line (made with intervals) then we can always find a finer cover C such that |C| ≤ 2( |C 1 | + . . . + |C k | ) . The blue cover has 3 rectangles. The red cover also has 3 rectangles. We can find a finer cover which has 9 rectangles.

How do intervals and rectangles differ? The blue cover has 3 intervals. The red cover also has 3 intervals. We can In general, if we have find a finer cover which has 5 intervals. C 1 , . . . , C k different covers of a compact line (made with intervals) then we can always find a finer cover C such that |C| ≤ 2( |C 1 | + . . . + |C k | ) . The blue cover has 3 rectangles. The red cover also has 3 rectangles. We In general, if you have can find a finer cover which has 9 rectangles. C 1 , . . . , C k different covers (made with rectangles) then we can always find a finer cover C such that |C| ≤ (2( |C 1 | + . . . + |C k | )) 2 .

We define the free-dimension of a compact space K using families C of finite covers C consisting of closed sets. We say that C is topologically cofinal if for every open cover of K there is a finer cover in C .

We define the free-dimension of a compact space K using families C of finite covers C consisting of closed sets. We say that C is topologically cofinal if for every open cover of K there is a finer cover in C . Definition (G.M.C. and G. Plebanek, 2018)

We define the free-dimension of a compact space K using families C of finite covers C consisting of closed sets. We say that C is topologically cofinal if for every open cover of K there is a finer cover in C . Definition (G.M.C. and G. Plebanek, 2018) Let d ∈ N ∪ { 0 } .

We define the free-dimension of a compact space K using families C of finite covers C consisting of closed sets. We say that C is topologically cofinal if for every open cover of K there is a finer cover in C . Definition (G.M.C. and G. Plebanek, 2018) Let d ∈ N ∪ { 0 } . We say that a compact space K has free dimension ≤ d

We define the free-dimension of a compact space K using families C of finite covers C consisting of closed sets. We say that C is topologically cofinal if for every open cover of K there is a finer cover in C . Definition (G.M.C. and G. Plebanek, 2018) Let d ∈ N ∪ { 0 } . We say that a compact space K has free dimension ≤ d ( free - dim ( K ) ≤ d)

We define the free-dimension of a compact space K using families C of finite covers C consisting of closed sets. We say that C is topologically cofinal if for every open cover of K there is a finer cover in C . Definition (G.M.C. and G. Plebanek, 2018) Let d ∈ N ∪ { 0 } . We say that a compact space K has free dimension ≤ d ( free - dim ( K ) ≤ d) if there are

We define the free-dimension of a compact space K using families C of finite covers C consisting of closed sets. We say that C is topologically cofinal if for every open cover of K there is a finer cover in C . Definition (G.M.C. and G. Plebanek, 2018) Let d ∈ N ∪ { 0 } . We say that a compact space K has free dimension ≤ d ( free - dim ( K ) ≤ d) if there are a topologically cofinal family C of finite closed covers, a constant M > 0

We define the free-dimension of a compact space K using families C of finite covers C consisting of closed sets. We say that C is topologically cofinal if for every open cover of K there is a finer cover in C . Definition (G.M.C. and G. Plebanek, 2018) Let d ∈ N ∪ { 0 } . We say that a compact space K has free dimension ≤ d ( free - dim ( K ) ≤ d) if there are a topologically cofinal family C of finite closed covers, a constant M > 0 and a function χ : C → N

We define the free-dimension of a compact space K using families C of finite covers C consisting of closed sets. We say that C is topologically cofinal if for every open cover of K there is a finer cover in C . Definition (G.M.C. and G. Plebanek, 2018) Let d ∈ N ∪ { 0 } . We say that a compact space K has free dimension ≤ d ( free - dim ( K ) ≤ d) if there are a topologically cofinal family C of finite closed covers, a constant M > 0 and a function χ : C → N such that for every k ∈ N and every C 1 , . . . , C k ∈ C there is a finer cover C such that |C| ≤ M ( χ ( C 1 ) + . . . + χ ( C k )) d .

We define the free-dimension of a compact space K using families C of finite covers C consisting of closed sets. We say that C is topologically cofinal if for every open cover of K there is a finer cover in C . Definition (G.M.C. and G. Plebanek, 2018) Let d ∈ N ∪ { 0 } . We say that a compact space K has free dimension ≤ d ( free - dim ( K ) ≤ d) if there are a topologically cofinal family C of finite closed covers, a constant M > 0 and a function χ : C → N such that for every k ∈ N and every C 1 , . . . , C k ∈ C there is a finer cover C such that |C| ≤ M ( χ ( C 1 ) + . . . + χ ( C k )) d . With this definition it is easy to check that free - dim ( L ) ≤ 1 if L is a compact line,

We define the free-dimension of a compact space K using families C of finite covers C consisting of closed sets. We say that C is topologically cofinal if for every open cover of K there is a finer cover in C . Definition (G.M.C. and G. Plebanek, 2018) Let d ∈ N ∪ { 0 } . We say that a compact space K has free dimension ≤ d ( free - dim ( K ) ≤ d) if there are a topologically cofinal family C of finite closed covers, a constant M > 0 and a function χ : C → N such that for every k ∈ N and every C 1 , . . . , C k ∈ C there is a finer cover C such that |C| ≤ M ( χ ( C 1 ) + . . . + χ ( C k )) d . With this definition it is easy to check that free - dim ( L ) ≤ 1 if L is a compact line, that free - dim ( K 1 ) ≤ free - dim ( K 2 ) if K 1 ⊆ K 2 or if K 1 is a continuous image of K 2 ,

We define the free-dimension of a compact space K using families C of finite covers C consisting of closed sets. We say that C is topologically cofinal if for every open cover of K there is a finer cover in C . Definition (G.M.C. and G. Plebanek, 2018) Let d ∈ N ∪ { 0 } . We say that a compact space K has free dimension ≤ d ( free - dim ( K ) ≤ d) if there are a topologically cofinal family C of finite closed covers, a constant M > 0 and a function χ : C → N such that for every k ∈ N and every C 1 , . . . , C k ∈ C there is a finer cover C such that |C| ≤ M ( χ ( C 1 ) + . . . + χ ( C k )) d . With this definition it is easy to check that free - dim ( L ) ≤ 1 if L is a compact line, that free - dim ( K 1 ) ≤ free - dim ( K 2 ) if K 1 ⊆ K 2 or if K 1 is a continuous image of K 2 , that free - dim ( K ) ≤ 1 if K is a metric compact space

We define the free-dimension of a compact space K using families C of finite covers C consisting of closed sets. We say that C is topologically cofinal if for every open cover of K there is a finer cover in C . Definition (G.M.C. and G. Plebanek, 2018) Let d ∈ N ∪ { 0 } . We say that a compact space K has free dimension ≤ d ( free - dim ( K ) ≤ d) if there are a topologically cofinal family C of finite closed covers, a constant M > 0 and a function χ : C → N such that for every k ∈ N and every C 1 , . . . , C k ∈ C there is a finer cover C such that |C| ≤ M ( χ ( C 1 ) + . . . + χ ( C k )) d . With this definition it is easy to check that free - dim ( L ) ≤ 1 if L is a compact line, that free - dim ( K 1 ) ≤ free - dim ( K 2 ) if K 1 ⊆ K 2 or if K 1 is a continuous image of K 2 , that free - dim ( K ) ≤ 1 if K is a metric compact space and that free - dim ( K 1 × . . . × K d ) ≤ free - dim ( K 1 ) + . . . + free - dim ( K d ).

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 .

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1]

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 .

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Main ingredients.

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Main ingredients. Since K is not metrizable, no countable family of functions separates points.

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Main ingredients. Since K is not metrizable, no countable family of functions separates points. Using this fact we can construct a family F = { f α : α < ω 1 } of continuous functions and points x 0 α , x 1 α ∈ K such that

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Main ingredients. Since K is not metrizable, no countable family of functions separates points. Using this fact we can construct a family F = { f α : α < ω 1 } of continuous functions and points x 0 α , x 1 α ∈ K such that 1 f α ( x 0 α ) = 0 and f α ( x 1 α ) = 1 for every α < ω 1 ;

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Main ingredients. Since K is not metrizable, no countable family of functions separates points. Using this fact we can construct a family F = { f α : α < ω 1 } of continuous functions and points x 0 α , x 1 α ∈ K such that 1 f α ( x 0 α ) = 0 and f α ( x 1 α ) = 1 for every α < ω 1 ; 2 f β ( x 0 α ) = f β ( x 1 α ) for every β < α < ω 1 .

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Main ingredients. Since K is not metrizable, no countable family of functions separates points. Using this fact we can construct a family F = { f α : α < ω 1 } of continuous functions and points x 0 α , x 1 α ∈ K such that 1 f α ( x 0 α ) = 0 and f α ( x 1 α ) = 1 for every α < ω 1 ; 2 f β ( x 0 α ) = f β ( x 1 α ) for every β < α < ω 1 . To prove the property stated in the Lemma use Ramsey.

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 .

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Sketch of the proof. Set K = K 1 × . . . × K d +1 . Suppose free - dim ( K ) ≤ d . Set C a topologically cofinal family of finite closed covers and χ : C → N such that for every C 1 , . . . , C k ∈ C there is a finer cover C with |C| ≤ ( χ ( C 1 ) + . . . + χ ( C k )) d .

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Sketch of the proof. Set K = K 1 × . . . × K d +1 . Suppose free - dim ( K ) ≤ d . Set C a topologically cofinal family of finite closed covers and χ : C → N such that for every C 1 , . . . , C k ∈ C there is a finer cover C with |C| ≤ ( χ ( C 1 ) + . . . + χ ( C k )) d . Notice that for every continuous function f ∈ C ( K ) there is a cover C f ∈ C such that Osc( f , C ) ≤ 1 3 for every C ∈ C f .

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Sketch of the proof. Set K = K 1 × . . . × K d +1 . Suppose free - dim ( K ) ≤ d . Set C a topologically cofinal family of finite closed covers and χ : C → N such that for every C 1 , . . . , C k ∈ C there is a finer cover C with |C| ≤ ( χ ( C 1 ) + . . . + χ ( C k )) d . Notice that for every continuous function f ∈ C ( K ) there is a cover C f ∈ C such that Osc( f , C ) ≤ 1 3 for every C ∈ C f . Take families F i ⊂ C ( K i ) as in the Lemma for every i ≤ d .

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Sketch of the proof. Set K = K 1 × . . . × K d +1 . Suppose free - dim ( K ) ≤ d . Set C a topologically cofinal family of finite closed covers and χ : C → N such that for every C 1 , . . . , C k ∈ C there is a finer cover C with |C| ≤ ( χ ( C 1 ) + . . . + χ ( C k )) d . Notice that for every continuous function f ∈ C ( K ) there is a cover C f ∈ C such that Osc( f , C ) ≤ 1 3 for every C ∈ C f . Take families F i ⊂ C ( K i ) as in the Lemma for every i ≤ d . WLOG, there is m ∈ N such that χ ( C f ) ≤ m for every f ∈ F 1 ∪ . . . F d .

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Sketch of the proof. Set K = K 1 × . . . × K d +1 . Suppose free - dim ( K ) ≤ d . Set C a topologically cofinal family of finite closed covers and χ : C → N such that for every C 1 , . . . , C k ∈ C there is a finer cover C with |C| ≤ ( χ ( C 1 ) + . . . + χ ( C k )) d . Notice that for every continuous function f ∈ C ( K ) there is a cover C f ∈ C such that Osc( f , C ) ≤ 1 3 for every C ∈ C f . Take families F i ⊂ C ( K i ) as in the Lemma for every i ≤ d . WLOG, there is m ∈ N such that χ ( C f ) ≤ m for every f ∈ F 1 ∪ . . . F d . Now notice that for any finite families G ⊂ F 1 ∪ . . . F d and H ⊂ C ( K d +1 ) there is a finite closed cover C such � d ≤ � d , with �� � that |C| ≤ f ∈G χ ( C f ) + � f ∈H χ ( C f ) |G| m + � f ∈H χ ( C f )

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Sketch of the proof. Set K = K 1 × . . . × K d +1 . Suppose free - dim ( K ) ≤ d . Set C a topologically cofinal family of finite closed covers and χ : C → N such that for every C 1 , . . . , C k ∈ C there is a finer cover C with |C| ≤ ( χ ( C 1 ) + . . . + χ ( C k )) d . Notice that for every continuous function f ∈ C ( K ) there is a cover C f ∈ C such that Osc( f , C ) ≤ 1 3 for every C ∈ C f . Take families F i ⊂ C ( K i ) as in the Lemma for every i ≤ d . WLOG, there is m ∈ N such that χ ( C f ) ≤ m for every f ∈ F 1 ∪ . . . F d . Now notice that for any finite families G ⊂ F 1 ∪ . . . F d and H ⊂ C ( K d +1 ) there is a finite closed cover C such � d ≤ � d , with �� f ∈G χ ( C f ) + � � |G| m + � that |C| ≤ f ∈H χ ( C f ) f ∈H χ ( C f ) C finer than every C f with f ∈ G ∪ H . This yields to a contradiction.

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Sketch of the proof. Set K = K 1 × . . . × K d +1 . Suppose free - dim ( K ) ≤ d . Set C a topologically cofinal family of finite closed covers and χ : C → N such that for every C 1 , . . . , C k ∈ C there is a finer cover C with |C| ≤ ( χ ( C 1 ) + . . . + χ ( C k )) d . Notice that for every continuous function f ∈ C ( K ) there is a cover C f ∈ C such that Osc( f , C ) ≤ 1 3 for every C ∈ C f . Take families F i ⊂ C ( K i ) as in the Lemma for every i ≤ d . WLOG, there is m ∈ N such that χ ( C f ) ≤ m for every f ∈ F 1 ∪ . . . F d . Now notice that for any finite families G ⊂ F 1 ∪ . . . F d and H ⊂ C ( K d +1 ) there is a finite closed cover C such � d ≤ � d , with �� f ∈G χ ( C f ) + � � |G| m + � that |C| ≤ f ∈H χ ( C f ) f ∈H χ ( C f ) C finer than every C f with f ∈ G ∪ H . This yields to a contradiction.

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Sketch of the proof. Set K = K 1 × . . . × K d +1 . Suppose free - dim ( K ) ≤ d . Set C a topologically cofinal family of finite closed covers and χ : C → N such that for every C 1 , . . . , C k ∈ C there is a finer cover C with |C| ≤ ( χ ( C 1 ) + . . . + χ ( C k )) d . Notice that for every continuous function f ∈ C ( K ) there is a cover C f ∈ C such that Osc( f , C ) ≤ 1 3 for every C ∈ C f . Take families F i ⊂ C ( K i ) as in the Lemma for every i ≤ d . WLOG, there is m ∈ N such that χ ( C f ) ≤ m for every f ∈ F 1 ∪ . . . F d . Now notice that for any finite families G ⊂ F 1 ∪ . . . F d and H ⊂ C ( K d +1 ) there is a finite closed cover C such � d ≤ � d , with �� f ∈G χ ( C f ) + � � |G| m + � that |C| ≤ f ∈H χ ( C f ) f ∈H χ ( C f ) C finer than every C f with f ∈ G ∪ H . This yields to a contradiction.

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Sketch of the proof. Set K = K 1 × . . . × K d +1 . Suppose free - dim ( K ) ≤ d . Set C a topologically cofinal family of finite closed covers and χ : C → N such that for every C 1 , . . . , C k ∈ C there is a finer cover C with |C| ≤ ( χ ( C 1 ) + . . . + χ ( C k )) d . Notice that for every continuous function f ∈ C ( K ) there is a cover C f ∈ C such that Osc( f , C ) ≤ 1 3 for every C ∈ C f . Take families F i ⊂ C ( K i ) as in the Lemma for every i ≤ d . WLOG, there is m ∈ N such that χ ( C f ) ≤ m for every f ∈ F 1 ∪ . . . F d . Now notice that for any finite families G ⊂ F 1 ∪ . . . F d and H ⊂ C ( K d +1 ) there is a finite closed cover C such � d ≤ � d , with �� � that |C| ≤ f ∈G χ ( C f ) + � f ∈H χ ( C f ) |G| m + � f ∈H χ ( C f ) C finer than every C f with f ∈ G ∪ H . This yields to a contradiction.

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Sketch of the proof. Set K = K 1 × . . . × K d +1 . Suppose free - dim ( K ) ≤ d . Set C a topologically cofinal family of finite closed covers and χ : C → N such that for every C 1 , . . . , C k ∈ C there is a finer cover C with |C| ≤ ( χ ( C 1 ) + . . . + χ ( C k )) d . Notice that for every continuous function f ∈ C ( K ) there is a cover C f ∈ C such that Osc( f , C ) ≤ 1 3 for every C ∈ C f . Take families F i ⊂ C ( K i ) as in the Lemma for every i ≤ d . WLOG, there is m ∈ N such that χ ( C f ) ≤ m for every f ∈ F 1 ∪ . . . F d . Now notice that for any finite families G ⊂ F 1 ∪ . . . F d and H ⊂ C ( K d +1 ) there is a finite closed cover C such � d ≤ � d , with �� � that |C| ≤ f ∈G χ ( C f ) + � f ∈H χ ( C f ) |G| m + � f ∈H χ ( C f ) C finer than every C f with f ∈ G ∪ H . This yields to a contradiction.

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Sketch of the proof. Set K = K 1 × . . . × K d +1 . Suppose free - dim ( K ) ≤ d . Set C a topologically cofinal family of finite closed covers and χ : C → N such that for every C 1 , . . . , C k ∈ C there is a finer cover C with |C| ≤ ( χ ( C 1 ) + . . . + χ ( C k )) d . Notice that for every continuous function f ∈ C ( K ) there is a cover C f ∈ C such that Osc( f , C ) ≤ 1 3 for every C ∈ C f . Take families F i ⊂ C ( K i ) as in the Lemma for every i ≤ d . WLOG, there is m ∈ N such that χ ( C f ) ≤ m for every f ∈ F 1 ∪ . . . F d . Now notice that for any finite families G ⊂ F 1 ∪ . . . F d and H ⊂ C ( K d +1 ) there is a finite closed cover C such � d ≤ � d , with �� f ∈G χ ( C f ) + � � |G| m + � that |C| ≤ f ∈H χ ( C f ) f ∈H χ ( C f ) C finer than every C f with f ∈ G ∪ H . This yields to a contradiction.

Theorem (G.M.C. and G. Plebanek,2018) If K 1 , . . . , K d are nonmetrizable compact spaces and K d +1 is an infinite compact space, then free - dim ( K 1 × . . . × K d +1 ) ≥ d + 1 . Lemma (Key Lemma) If K is a nonmetrizable compact space, then there is an uncountable family F of continuous functions f : K → [0 , 1] such that for every infinite family F ′ ⊆ F and for every n ∈ N there are functions f 1 , . . . , f n ∈ F ′ and points x 1 , . . . , x n +1 ∈ K such that for every x j � = x j ′ there is k ≤ n such that | f k ( x j ) − f k ( x j ′ ) | ≥ 1 2 . Sketch of the proof. Set K = K 1 × . . . × K d +1 . Suppose free - dim ( K ) ≤ d . Set C a topologically cofinal family of finite closed covers and χ : C → N such that for every C 1 , . . . , C k ∈ C there is a finer cover C with |C| ≤ ( χ ( C 1 ) + . . . + χ ( C k )) d . Notice that for every continuous function f ∈ C ( K ) there is a cover C f ∈ C such that Osc( f , C ) ≤ 1 3 for every C ∈ C f . Take families F i ⊂ C ( K i ) as in the Lemma for every i ≤ d . WLOG, there is m ∈ N such that χ ( C f ) ≤ m for every f ∈ F 1 ∪ . . . F d . Now notice that for any finite families G ⊂ F 1 ∪ . . . F d and H ⊂ C ( K d +1 ) there is a finite closed cover C such � d ≤ � d , with �� � that |C| ≤ f ∈G χ ( C f ) + � f ∈H χ ( C f ) |G| m + � f ∈H χ ( C f ) C finer than every C f with f ∈ G ∪ H . This yields to a contradiction.

Looking for the free dimension of a Banach Space

Is it possible to define the free dimension of a Banach space in such a way that 1 free - dim ( Y ) ≤ free - dim ( X ) whenever Y is a subspace of X

Is it possible to define the free dimension of a Banach space in such a way that 1 free - dim ( Y ) ≤ free - dim ( X ) whenever Y is a subspace of X or Y is a quotient of X ;

Is it possible to define the free dimension of a Banach space in such a way that 1 free - dim ( Y ) ≤ free - dim ( X ) whenever Y is a subspace of X or Y is a quotient of X ; 2 free - dim ( C ( L 1 × . . . × L d )) ≤ d for every compact lines L 1 , . . . , L d ;