Chapter 2: Topological Spaces

Now that we have had some experience with metric spaces, we should be ready for the more general notion of a topological space.

BEGIN WEEK 3 (3/14-3/20)

Topological Spaces
Definition 2.1: Let $$X$$ be a set, and suppose $$\mathcal{T}$$ is a collection of subsets of $$X$$ with the following properties:

i) $$\emptyset \in \mathcal{T}$$;

ii) $$X \in \mathcal{T}$$;

iii) The union of any collection of members of $$\mathcal{T}$$ is also a member of $$\mathcal{T}$$;

iv) The intersection of any two members of $$\mathcal{T}$$ is also a member of $$\mathcal{T}$$.

Then we say that the pair $$(X,\mathcal{T})$$ is a topological space and $$\mathcal{T}$$ is the topology on $$X$$. (Often, when the topology $$\mathcal{T}$$ is understood, we simply refer to the topological space as $$X$$.) The members of $$\mathcal{T}$$ are called open sets.

Exercise 2.2: Let $$X = \{a,b,c\}$$ and let $$\mathcal{T} = \{\emptyset, \{a,b,c\}, \{a\}, \{b,c\}\}$$. Verify that $$\mathcal{T}$$ is a topology on $$X$$. (Hence, $$(X,\mathcal{T})$$ is a topological space.)

Exercise 2.3: Let $$X = \{1,2,3\}$$ and let $$\mathcal{T} = \{ \emptyset, \{1,2,3\}, \{1\}, \{2\}, \{2,3\} \}$$. Show that $$\mathcal{T}$$ is NOT a topology on $$X$$.

Exercise 2.4: Let $$X$$ be any metric space with metric $$d$$. Show that $$X$$ is a topological space, where the topology $$\mathcal{T}$$ consists of $$\emptyset$$ and all subsets of $$X$$ that are open in the sense of Definition 1.7.

From now on, if our topological space is given a metric or is assumed to have a metric (such as the distance metric on $$\mathbb{R}^n$$), then we assume it has the induced topology from Exercise 2.4 unless stated otherwise.

Definition 2.5: Let $$X$$ be any set, and let $$\mathcal{T} = \mathcal{P}(X)$$, the set of all subsets of $$X$$ (known as the power set). Then $$(X,\mathcal{T})$$ is known said to have the discrete topology.

Exercise 2.6: Verify that the discrete topology is a topology.

Definition 2.7: Let $$X$$ be any set, and let $$\mathcal{T}= \{ \emptyset, X \}$$. Then $$\mathcal{T}$$ is called the indiscrete topology on $$X$$. (The indiscrete topology is the smallest possible topology on $$X$$.)

Exercise 2.8: Let $$X$$ be any set, and let $$\mathcal{T}$$ consist of the empty set and all subsets of $$X$$ whose complement in $$X$$ is finite. Show that $$\mathcal{T}$$ is a topology. (This is known as the cofinite topology.)

Open and Closed Sets
Definition 2.9: As stated before, if $$(X,\mathcal{T})$$ is a topological space, the members of the topology $$\mathcal{T}$$ are called the open sets of $$X$$. A subset of $$X$$ is called a closed set if its complement is an open set.

Example 2.10: Let $$X = \{1,2,3\}$$, and let $$T = \{\emptyset, X, \{1,2\}, \{2,3\}, \{2\}\}$$. Then because $$\{2\}$$ is an open set, its complement in $$X$$, $$\{1,3\}$$, is a closed set.

Definition 2.11: As in the metric setting, it is possible for a subset of $$X$$ to be neither open nor closed (e.g., in Example 2.2, $$\{a,b\}$$). It is also possible for a subset of $$X$$ to be both open and closed, which we call clopen. (E.g., if $$X$$ has the discrete topology, then every subset of $$X$$ is clopen).

Exercise 2.12: Let $$d$$ be the euclidean metric on $$\mathbb{R}^2$$. Find every clopen set in the Euclidean topology induced by that metric on $$\mathbb{R}^2$$.

Proposition 2.13: If $$X$$ is a topological space, then (i) the intersection of any collection of closed sets is a closed set; (ii) the union of finitely many closed sets is a closed set.

Exercise 2.14: Find an example to show that a union of infinitely many closed sets may not be closed.

Definition 2.15: Let $$A$$ be a subset of some topological space $$X$$. Then we say a point $$p$$ in $$X$$ is a limit point of $$A$$ if every open set in $$X$$ that contains $$p$$ also contains a point of $$A$$ different from $$p$$. (Note that a limit point of $$A$$ need not necessarily be an element of $$A$$.)

Example 2.16: Let $$X = \{a,b,c,d\}$$, and let $$T = \{\emptyset, X, \{a,b,c\}, \{a,b,d\}, \{a,b\}, \{c\} \}$$. Then $$b$$ is a limit point of the set $$\{a,c,d\}$$ because every open set that contains $$b$$ contains a point of $$\{a,c,d\}$$ different from $$b$$. On the other hand, $$c$$ is not a limit point of the set $$\{a,c,d\}$$ because there exists an open set containing $$c$$--namely, $$\{c\}$$--that contains no point in $$\{a,c,d\}$$ different from $$c$$.

Exercise 2.17.a: Consider the metric space $$\mathbb{R}$$ (with the usual Euclidean metric). Show that the point $$0$$ is a limit point of the set $$K = \{1/n : n\text{ is a positive integer }\}$$.

Exercise 2.17.b: Now show that every point in $$K$$ is NOT a limit point of $$K$$.

Proposition 2.18: Let $$X$$ be a topological space and let $$A$$ be a subset of $$X$$. Then $$A$$ is closed iff $$A$$ contains all its limit points.

END OF WEEK 3

Closure, Interior, and Boundary
Definition 2.19: If $$A$$ is a subset of $$X$$, we define $$A'$$ to be the set of all limit points of $$A$$ in $$X$$.

Definition 2.20: The closure of $$A$$, written $$Cl(A)$$ or $$\overline{A}$$, is $$A \cup A'$$.

Exercise 2.21: Let $$A=(0,1)=\{x\in \mathbb{R}:0 < x < 1 \}$$. Find $$A'$$ and $$\overline{A}$$.

Proposition 2.22: Let $$A \subseteq X$$, and $$\mathcal{C}(A)= \{ C : C \text{ is closed and } C \supseteq A\}$$. Then $$\overline{A}= \bigcap \mathcal{C}(A)$$, that is, $$\overline{A}$$ is the intersection of all closed supersets of $$A$$.

Corollary 2.23: $$\overline{A}$$ is the smallest closed set containing $$A$$; that is, $$\overline{A} \subseteq C$$ for every closed set $$C$$ containing $$A$$.

Corollary 2.24: A set $$K$$ is closed in $$X$$ if and only if $$K = \overline{K}$$.

Definition 2.25: Let $$A$$ be a subset of $$X$$. We say $$p$$ is an interior point of $$A$$ if there exists some open set $$U$$ so that $$p \in U \subseteq A$$.

Definition 2.26: The interior of $$A$$, written $$Int(A)$$, is the set of all interior points of $$A$$.

Proposition 2.27: Let $$A \subseteq X$$, and $$\mathcal{U}(A)= \{ U : U \text{ is open and } U \subseteq A\}$$. Show that $$Int(A)= \bigcup \mathcal{U}(A)$$, that is, $$int(A)$$ is the union of all open subsets of $$A$$.

Corollary 2.28: $$Int(A)$$ is the largest open set contained in $$A$$; that is, $$int(A) \supseteq U$$ for every open set $$U$$ within $$A$$.

Corollary 2.29: A set $$U$$ is closed in $$X$$ if and only if $$U = Int(U)$$.

Definition 2.30: Let $$A$$ be a subset of $$X$$. We say $$p$$ is a boundary point of $$A$$ if every open set containing $$p$$ contains a point in $$A$$ and a point in the complement of $$A$$.

Definition 2.31: The boundary of $$A$$, denoted $$Bd(A)$$, is the set of all boundary points of $$A$$.

Exercise 2.32.a: Consider $$A = (0,1]\subseteq \mathbb{R}$$. Find $$A',Cl(A),Int(A),Bd(A)$$.

Exercise 2.32.b: Give $$\omega=\{0,1,2,\dots\}$$ the cofinite topology. Let $$A=\{0,10,100,1000,1001,1002,1003,\dots\}$$. Find $$A',Cl(A),Int(A),Bd(A)$$.

We may characterize the interior, boundary, closure, and limit point sets in terms of each other. We assume $$A$$ is a subset of some topological space $$X$$. (Recall that for a given "universe", $$X$$ in this case, the complement of a set $$A \subseteq X$$ is $$A^c = X \setminus A$$.)

Proposition 2.33.a: $$Cl(A) = Int(A) \cup A'$$.

Proposition 2.33.b: $$Cl(A) = Int(A) \cup Bd(A)$$.

Proposition 2.33.c: $$Int(A) = A \setminus Bd(A)$$.

Proposition 2.33.d: $$Int(A) = (Cl(A^c))^c$$.

Proposition 2.33.e: $$Cl(A) = (Int(A^c))^c$$.

Proposition 2.33.f: $$Bd(A) = Cl(A) \setminus Int(A)$$.

Proposition 2.33.g: $$Bd(A) = Bd(A^c)$$.

We close by defining an "exterior" (which we will rarely use), and use it to partition any space into three parts based on a particular subset of the space.

Definition 2.34: Call $$x$$ an exterior point of $$A$$ if there exists an open set $$U$$ containing $$x$$ such that $$U \cap A = \emptyset$$.

Definition 2.35: The exterior of $$A$$, written $$Ext(A)$$, is the set of all exterior points of $$A$$.

Proposition 2.36: $$Ext(A) = Int(A^c)$$.

Theorem 2.37: For any subset $$A$$ of a topological space $$X$$, $$Int(A),Bd(A),Ext(A)$$ partition the space into three disjoint sets. That is, no two of $$Int(A),Bd(A),Ext(A)$$ have a nonempty intersection (they are pairwise disjoint) and $$X= Int(A)\cup Bd(A)\cup Ext(A)$$.

Sequences and Continuity
In Chapter 1, we stated that for a metric space, a sequence $$\{x_n\}$$ converges to $$x$$ iff for all $$\epsilon > 0$$, there exists some large enough integer $$N$$ so that if $$n \ge N$$, then $$x_n$$ lies within $$\epsilon$$ of $$x$$. However, for an arbitrary topological space $$X$$, there may not be a notion of “distance” in $$X$$ at all. Therefore, we need to define what it means for a sequence to converge in the more general setting of a topological space (without using any $$\epsilon$$ in our definition).

Definition 2.38: Let $$\{x_n\}$$ be a sequence of points in a topological space $$X$$. Then we say $$\{x_n\}$$ converges to $$x$$ in $$X$$ iff for every open subset $$U$$ of $$X$$ containing $$x$$, there exists some large enough integer $$N$$ so that $$x_n \in U$$ whenever $$n \ge N$$.

Proposition 2.39: The sequence $$\{x_n\}$$ converges to $$x$$ if and only if for any open set $$U$$ containing $$x$$, all but finitely many points in the sequence are contained in $$U$$.

We also need a definition of continuity that does not use a notion of distance, i.e., does not use any $$\epsilon$$'s or $$\delta$$'s.

Definition 2.40: Let $$(X,\mathcal{T}_{X})$$ and $$(Y, \mathcal{T}_{Y})$$ be topological spaces and let $$f: X \rightarrow Y$$ be a function. Then we say $$f$$ is continuous at $$x$$ if, for every open set $$V \subseteq Y$$ containing $$f(x)$$, there exists an open set $$U \subseteq X$$ containing $$x$$ with $$f(U) \subseteq V$$. (If $$f$$ is continuous at every point of $$X$$, we simply say $$f$$ is continuous.)

Proposition 2.40: Let $$X,Y$$ be topological spaces. Then $$f: X \rightarrow Y$$ is continuous if and only if $$f^{-1}(O)$$ is open in $$X$$ for every open $$O \subset Y$$.

The statement of Proposition 2.40 is like the statement of Proposition 1.21, but with $$X,Y$$ being arbitrary topological spaces instead of metric spaces. We also have an analog of Proposition 1.20:

Proposition 2.41: Let $$X,Y$$ be topological spaces. If $$f: X \rightarrow Y$$ is continuous and $$\{x_n\} \subset X$$ converges to $$x \in X$$, then $$\{f(x_n)\}$$ converges to $$f(x) \in Y$$.