College Math Teaching

April 10, 2015

Cantor sets and countable products of discrete spaces (0, 1)^Z

Filed under: advanced mathematics, analysis, point set topology, sequences — Tags: , , , — collegemathteaching @ 11:09 am

This might seem like a strange topic but right now our topology class is studying compact metric spaces. One of the more famous of these is the “Cantor set” or “Cantor space”. I discussed the basics of these here.

Now if you know the relationship between a countable product of two point discrete spaces (in the product topology) and Cantor spaces/Cantor Sets, this post is probably too elementary for you.

Construction: start with a two point set D = \{0, 2 \} and give it the discrete topology. The reason for choosing 0 and 2 to represent the elements will become clear later. Of course, D_2 is a compact metric space (with the discrete metric: d(x,y) = 1 if x \neq y .

Now consider the infinite product of such spaces with the product topology: C = \Pi^{\infty}_{i=1} D_i where each D_i is homeomorphic to D . It follows from the Tychonoff Theorem that C is compact, though we can prove this directly: Let \cup_{\alpha \in I} U_{\alpha} be any open cover for C . Then choose an arbitrary U from this open cover; because we are using the product topology U = O_1 \times O_2 \times ....O_k \times (\Pi_{i=k+1}^{\infty} D_i ) where each O_i is a one or two point set. This means that the cardinality of C - U is at most 2^k -1 which requires at most 2^k -1 elements of the open cover to cover.

Now let’s examine some properties.

Clearly the space is Hausdorff (T_2 ) and uncountable.

1. Every point of C is a limit point of C . To see this: denote x \in C by the sequence \{x_i \} where x_i \in \{0, 2 \} . Then any open set containing \{ x_i \} is O_1 \times O_2 \times...O_k \times \Pi^{\infty}_{i=k+1} D_i and contains ALL points y_i where y_i = x_i for i = \{1, 2, ...k \} . So all points of C are accumulation points of C ; in fact they are condensation points (or perfect limit points ).

(refresher: accumulation points are those for which every open neighborhood contains an infinite number of points of the set in question; condensation points contain an uncountable number of points, and perfect limit points are those for which every open neighborhood contains as many points as the set in question has (same cardinality).

2. C is totally disconnected (the components are one point sets). Here is how we will show this: given x, y \in C, x \neq y, there exists disjoint open sets U_x, U_y, x \in U_x, y \in U_y, U_x \cup U_y = C . Proof of claim: if x \neq y there exists a first coordinate k for which x_k \neq y_k (that is, a first k for which the canonical projection maps disagree (\pi_k(x) \neq pi_k(y) ). Then
U_x = D_1 \times D_2 \times ....\times D_{k-1} \times x_k \times \Pi^{\infty}_{i=k+1} D_i,

U_y = D_1 \times D_2 \times.....\times D_{k-1} \times y_k \times \Pi^{\infty}_{i = k+1} D_i

are the required disjoint open sets whose union is all of C .

3. C is countable, as basis elements for open sets consist of finite sequences of 0’s and 2’s followed by an infinite product of D_i .

4. C is metrizable as well; d(x,y) = \sum^{\infty}_{i=1} \frac{|x_i - y_i|}{3^i} . Note that is metric is well defined. Suppose x \neq y . Then there is a first k, x_k \neq y_k . Then note

d(x,y) = \frac{|x_k - y_k|}{3^k} + \sum^{\infty}_{i = k+1} \frac{|x_i - y_i|}{3^i} \rightarrow |x_k -y_k| =2 = \sum^{\infty}_{i=1} \frac{|x_{i+k} -y_{i+k}|}{3^i} \leq \frac{1}{3} \frac{1}{1 -\frac{2}{3}} =1

which is impossible.

5. By construction C is uncountable, though this follows from the fact that C is compact, Haudorff and dense in itself.

6. C \times C is homeomorphic to C . The homeomorphism is given by f( \{x_i \}, \{y_i \}) = \{ x_1, y_1, x_2, y_2,... \} \in C . It follows that C is homeomorphic to a finite product with itself (product topology). Here we use the fact that if f: X \rightarrow Y is a continuous bijection with X compact and Y Hausdorff then f is a homeomorphism.

Now we can say a bit more: if C_i is a copy of C then \Pi^{\infty}_{i =1 } C_i is homeomorphic to C . This will follow from subsequent work, but we can prove this right now, provided we review some basic facts about countable products and counting.

First lets show that there is a bijection between Z \times Z and Z . A bijection is suggested by this diagram:


which has the following formula (coming up with it is fun; it uses the fact that \sum^k _{n=1} n = \frac{k(k+1)}{2} :

\phi(k,1) = \frac{(k)(k+1)}{2} for k even
\phi(k,1) = \frac{(k-1)(k)}{2} + 1 for k odd
\phi(k-j, j+1) =\phi(k,1) + j for k odd, j \in \{1, 2, ...k-1 \}
\phi(k-j, j+1) = \phi(k,1) - j for k even, j \in \{1, 2, ...k-1 \}

Here is a different bijection; it is a fun exercise to come up with the relevant formulas:


Now lets give the map between \Pi^{\infty}_{i=1} C_i and C . Let \{ y_i \} \in C and denote the elements of \Pi^{\infty}_{i=1} C_i by \{ x^i_j \} where \{ x_1^1, x_2^1, x_3^ 1....\} \in C_1, \{x_1^2, x_2 ^2, x_3^3, ....\} \in C_2, ....\{x_1^k, x_2^k, .....\} \in C_k ... .

We now describe a map f: C \rightarrow \Pi^{\infty}_{i=1} C_i by

f(\{y_i \}) = \{ x^i_j \} = \{y_{\phi(i,j)} \}

Example: x^1_1 = y_1, x^1_2 = y_2, x^2_1 = y_3, x^3_1 =y_4, x^2_2 = y_5, x^1_3 =y_6,...

That this is a bijection between compact Hausdorff spaces is immediate. If we show that f^{-1} is continuous, we will have shown that f is a homeomorphism.

But that isn’t hard to do. Let U \subset C be open; U = U_1 \times U_2 \times U_3.... \times U_{m-1} \times \Pi^{\infty}_{k=m} C_k . Then there is some k_m for which \phi(k_m, 1) \geq M . Then if f^i_j denotes the i,j component of f we wee that for all i+j \geq k_m+1, f^i_j(U) = C (these are entries on or below the diagonal containing (k,1) depending on whether k_m is even or odd.

So f(U) is of the form V_1 \times V_2 \times ....V_{k_m} \times \Pi^{\infty}_{i = k_m +1} C_i where each V_j is open in C_j . This is an open set in the product topology of \Pi^{\infty}_{i=1} C_i so this shows that f^{-1} is continuous. Therefore f^{-1} is a homeomorphism, therefore so is f.

Ok, what does this have to do with Cantor Sets and Cantor Spaces?

If you know what the “middle thirds” Cantor Set is I urge you stop reading now and prove that that Cantor set is indeed homeomorphic to C as we have described it. I’ll give this quote from Willard, page 121 (Hardback edition), section 17.9 in Chapter 6:

The proof is left as an exercise. You should do it if you think you can’t, since it will teach you a lot about product spaces.

What I will do I’ll give a geometric description of a Cantor set and show that this description, which easily includes the “deleted interval” Cantor sets that are used in analysis courses, is homeomorphic to C .

Set up
I’ll call this set F and describe it as follows:

F \subset R^n (for those interested in the topology of manifolds this poses no restrictions since any manifold embeds in R^n for sufficiently high n ).

Reminder: the diameter of a set F \subset R^n will be lub \{ d(x,y) | x, y \in F \}
Let \epsilon_1, \epsilon_2, \epsilon_3 .....\epsilon_k ... be a strictly decreasing sequence of positive real numbers such that \epsilon_k \rightarrow 0 .

Let F^0 be some closed n-ball in R^n (that is, F^) is a subset homeomorphic to a closed n-ball; we will use that convention throughout)

Let F^1_{(0) }, F^1_{(2)} be two disjoint closed n-balls in the interior of F^0 , each of which has diameter less than \epsilon_1 .

F^1 = F^1_{(0) } \cup F^1_{(2)}

Let F^2_{(0, 0)}, F^2_{(0,2)} be disjoint closed n-balls in the interior F^1_{(0) } and F^2_{(2,0)}, F^2_{(2,2)} be disjoint closed n-balls in the interior of F^1_{(2) } , each of which (all 4 balls) have diameter less that \epsilon_2 . Let F^2 = F^2_{(0, 0)}\cup F^2_{(0,2)} \cup F^2_{(2, 0)} \cup F^2_{(2,2)}


To describe the construction inductively we will use a bit of notation: a_i \in \{0, 2 \} for all i \in \{1, 2, ...\} and \{a_i \} will represent an infinite sequence of such a_i .
Now if F^k has been defined, we let F^{k+1}_{(a_1, a_2, ...a_{k}, 0)} and F^{k+1}_{(a_1, a_2,....,a_{k}, 2)} be disjoint closed n-balls of diameter less than \epsilon_{k+1} which lie in the interior of F^k_{(a_1, a_2,....a_k) } . Note that F^{k+1} consists of 2^{k+1} disjoint closed n-balls.

Now let F = \cap^{\infty}_{i=1} F^i . Since these are compact sets with the finite intersection property (\cap^{m}_{i=1}F^i =F^i \neq \emptyset for all m ), F is non empty and compact. Now for any choice of sequence \{a_i \} we have F_{ \{a_i \} } =\cap^{\infty}_{i=1} F^i_{(a_1, ...a_i)} is nonempty by the finite intersection property. On the other hand, if x, y \in F, x \neq y then d(x,y) = \delta > 0 so choose \epsilon_m such that \epsilon_m < \delta . Then x, y lie in different components of F^m since the diameters of these components are less than \epsilon_m .

Then we can say that the F_{ \{a_i} \} uniquely define the points of F . We can call such points x_{ \{a_i \} }

Note: in the subspace topology, the F^k_{(a_1, a_2, ...a_k)} are open sets, as well as being closed.

Finding a homeomorphism from F to C .
Let f: F \rightarrow C be defined by f( x_{ \{a_i \} } ) = \{a_i \} . This is a bijection. To show continuity: consider the open set U =  y_1 \times y_2 ....\times y_m \times \Pi^{\infty}_{i=m} D_i . Under f this pulls back to the open set (in the subspace topology) F^{m+1}_{(y1, y2, ...y_m, 0 ) } \cup F^{m+1}_{(y1, y2, ...y_m, 2)} hence f is continuous. Because F is compact and C is Hausdorff, f is a homeomorphism.

This ends part I.

We have shown that the Cantor sets defined geometrically and defined via “deleted intervals” are homeomorphic to C . What we have not shown is the following:

Let X be a compact Hausdorff space which is dense in itself (every point is a limit point) and totally disconnected (components are one point sets). Then X is homeomorphic to C . That will be part II.


March 15, 2015

Compact spaces and Tychonoff’s Theorem I

Note to the reader: when I first started this post, I thought it would be a few easy paragraphs. The size began to swell, so I am taking this post in several bite sized parts. This is part I.

Pretty much everyone knows what a compact space is. But not everyone is up on equivalent definitions and on how to prove that the arbitrary product of compact spaces is compact.
The genesis of this blog post is David Wright’s Proceedings of the American Mathematical Society paper (1994) on Tychonoff’s Theorem.

Since I am also writing for my undergraduate topology class, I’ll keep things elementary where possible and perhaps put in more detail than a professional mathematician would have patience for.

I should start by saying why this topic is dear to me: my research area is knot theory; in particular I studied embeddings of the circle into the 3-dimensional sphere S^3 , which can be thought of as the “compactification” of R^3 ; basically one starts with R^3 and then adds a point \infty and declares that the neighborhoods of this new point will be generated by sets of the following form: \{ (x,y,z) | x^2 + y^2 + z^2 > M^2 \} The reason we do this: we often study the complement of the embedded circle, and it is desirable to have a compact set as the complement.

I’ve also studied (in detail) certain classes of embeddings of the real line into non-compact manifolds; to make this study a bit more focused, I insist that such embeddings be “proper” in that the inverse image of a compact set be compact. Hence compactness comes up again, even when the objects of study are not compact.

So, what do we mean by “compact”?

Instead of just blurting out the definition and equivalent formulations, I’ll try to give some intuition. If we are talking about a subset of a metric space, a compact subset is one that is both closed and bounded. Now that is NOT the definition of compactness, though it is true that:

Given a set X \subset R^n , X is compact if and only if X is both closed (as a topological subset) and bounded (in that it fits within a sufficiently large closed ball). In R^1 compact subsets can be thought of as selected finite unions and arbitrary intersections of closed intervals. In the higher dimensions, think of the finite union and arbitrary intersections of things like closed balls.

Now it is true that if f:X \rightarrow Y is continuous, then if X is a compact topological space, then f(X) is compact (either as a space, or in the subspace topology, if f is not onto.

This leads to a big theorem of calculus: the Extreme Value Theorem: if f:R^n \rightarrow R is continuous over a compact subset D \subset R^n then f attains both a maximum and a minimum value over D .

Now in calculus, we rarely use the word “compact” but instead say something about D be a closed, bounded subset. In the case where n = 1 we usually say that D =[a,b] , a closed interval.

So, in terms of intuition, if one is thinking about subsets of R^n , one can think of a compact space as a domain on which any continuous real valued function always attains both a minimum and a maximum value.

Now for the definition
We need some terminology: a collection of open sets U_{\alpha} is said to be an open cover of a space X if \cup_{\beta \in I } U_{\beta} = X and if A \subset X a collection of open sets is said to be an open cover of A if A \subset \cup_{\beta \in I } U_{\beta} A finite subcover is a finite subcollection of the open sets such that \cup^k_{i=1} U_i = \cup_{\beta \in I} U_{\beta} .

Here is an example: (\frac{3}{4}, 1] \cup^{\infty}_{n=1} [0, \frac{n}{n+1}) is an open cover of [0,1] in the subspace topology. A finite subcover (from this collection) would be [0, \frac{4}{5}) \cup (\frac{3}{4}, 1]

Let X be a topological space. We say that X is a compact topological space if any open over of X has a finite subcover. If C \subset X we say that C is a compact subset of X if any open cover of C has a finite subcover.

Prior to going through examples, I think that it is wise to mention something. One logically equivalent definition is this: A space (or a subset) is compact if every cover by open basis elements has a finite subcover. Here is why: if X is compact, then ANY open cover has a finite subcover, and an open cover by basis elements is an open cover. On the other hand: if we assume the “every open cover by open basis elements has a finite subcover” condition: then if \mathscr{U} is an open cover, then we can view this open cover as an open cover of the basis elements whose union is each open U_{\beta} \in \mathscr{U} . This open cover of basis elements has a finite subcover of basis elements..say B_1, B_2, ....B_k . Then for each basis element, choose a single U_i \in \mathscr{U} for which B_i \subset U_i . That is the required open subcover.

Now, when convenient, we can assume that the open cover in question (during a proof) consists of basic open sets. That will simplify things at times.

So, what are some compact spaces and sets, and what are some basic facts?

Let’s see some compact sets, some non compact sets and see some basic facts.

1. Let X be any topological space and A \subset X a finite subset. Then A is a compact subset. Proof: given any open cover of A choose one open set per element of A which contains said element.

2. Let R have the usual topology. Then the integers Z \subset R^1 is not a compact subset; choose the open cover \cup^{\infty}_{n = -\infty} (n - \frac{1}{4}, n+ \frac{1}{4}) is an infinite cover with no finite subcover. In fact, ANY unbounded subset A \subset R^n in the usual metric topology fails to be compact: for a \in A with d(a, 0) \geq n choose B_a(\frac{1}{n}) ; clearly this open cover can have no finite subcover.

3. The finite union of compact subsets is compact (easy exercise).

4. If C \subset X is compact and X is a Hausdorff topological space (T_2 ) then C is closed. Here is why: let x \notin C and for every c \in C choose U_c, V_c open where x \in U_c, c \in V_c . Now \cup_{c \in C}V_c is an open set which contains C and has a finite subcover \cup_{i=1}^k V_i Note that each U_i is an open set which contains x and now we have only a finite number of these. Hence x \in \cap^k_{i=1} U_i which is disjoint from \cup_{i=1}^k V_i which contains C . Because x was an arbitrary point in X -C , X-C is open which means that C is closed. Note: this proof, with one minor addition, shows that a compact Hausdorff space is regular (T_3 ) we need only show that a closed subset of a compact Hausdorff space is compact. That is easy enough to do: let \mathscr{U} be an open cover for C ; then the collection \mathscr{U} \cup (X-C) is an open cover for X , which has a finite subcover. Let that be \cup^k_{i=1} U_i \cup (X-C) where each U_i \in \mathscr{U} . Now since X-C does not cover C, \cup^k_{i=1} U_i does.

So we have proved that a closed subset of a compact set is compact.

5. Let R (or any infinite set) be given the finite complement topology (that is, the open sets are the empty set together with sets whose complements consist of a finite number of points). Then ANY subset is compact! Here is why: let C be any set and let \mathscr{U} be any open cover. Choose U_1 \in \mathscr{U}. Since X -U_1 is a finite set of points, only a finite number of them can be in C , say c_1, c_2, ...c_k . Then for each of these, select one open set in the open cover that contains the point; that is the finite subcover.

Note: this shows that non-closed sets can be compact sets, if the underlying topology is not Hausdorff.

6. If f: X \rightarrow Y is continuous and onto and X is compact, then so is Y . Proof: let \cup_{\beta \in I} U_{\beta} cover Y and note that \cup_{\beta}f^{-1}(U_{\beta}) covers X , hence a finite number of these open sets cover: X = \cup^{k}_{i=1}f^{-1}(U_i). Therefore \cup^k_{i=1}U_i covers Y . Note: this shows that being compact is a topological invariant; that is, if two spaces are homeomorphic, then either both spaces are compact or neither one is.

7. Ok, let’s finally prove something. Let R^1 have the usual topology. Then [0, 1] (and therefore any closed interval) is compact. This is (part) of the famous Heine-Borel Theorem. The proof uses the least upper bound axiom of the real numbers.

Let \mathscr{U} be any open cover for [0,1] . If no finite subcover exists, let x be the least upper bound of the subset F of [0,1] that CAN be covered by a finite subcollection of \mathscr{U} . Now x > 0 because at least one element of \mathscr{U} contains 0 and therefore contains [0, \delta ) for some \delta > 0 . Assume that x < 1 . Now suppose x \in F , that is x is part of the subset that can be covered by a finite subcover. Then because x \in U_{\beta} for some U_{\beta} \in \mathscr{U} then (x-\delta, x + \delta) \subset U_{\beta} which means that x + \delta \in F , which means that x isn’t an upper bound for F .

Now suppose x \notin F ; then because x < 1 there is still some U_{\beta} where (x-\delta, x+ \delta) \subset U_{\beta} . But since x = lub(F) then x - \delta \in F and so [0, x- \delta ) \subset F . So if F can be covered by \cup^k_{i=1} U_i then \cup^k_{i=1} U_i \cup U_{\beta} is a finite subcover of [0, x + \delta ) which means that x was not an upper bound. It follows that x = 1 which means that the unit interval is compact.

Now what about the closed ball in R^n ? The traditional way is to show that the closed ball is a closed subset of a closed hypercube in R^n and so if we show that the product of compact spaces is compact, we would be done. That is for later.

8. Now endow R^1 with the lower limit topology. That is, the open sets are generated by basis elements [a, b) . Note that the lower limit topology is strictly finer than the usual topology. Now in this topology: [0,1] is not compact. (note: none of (0,1), [0,1), (0, 1] are compact in the coarser usual topology, so there is no need to consider these). To see this, cover [0,1] by \cup ^{\infty}_{n=1} [0, \frac{n}{n+1}) \cup [1, \frac{3}{2}) and it is easy to see that this open cover has no finite subcover. In fact, with a bit of work, one can show that every compact subset is at most countable and nowhere dense; in fact, if A is compact in the lower limit topology and a \in A there exists some y_a where (y_a, a) \cap A = \emptyset .

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