### Notes on Topology

This post is the note from reading Bert Mendelson’s book Introduction to Topology, 3rd Edition. At the moment, the note is obviously unfinished and I am still reading the book.

#### 1` `Metric Spaces

##### 1.1` `Definition of Metric Spaces

Definition (Metric Space): \langle X, d \rangle is a metric space if X is the underlying set, and d : X \times X \to \mathbb{R} is a distance function, which satisfies

\forall x, y \in X, d(x, y) \geq 0

\forall x, y \in X, d(x, y) = 0 \text{ only if } x = y

\forall x, y \in X, d(x, y) = d(y, x)

\forall x, y, z \in X, d(x, z) \leq d(x, y) + d(y, z)

The last one assert the transitivity of closeness: if x is close to y and y is close to z, then x is close to z.

Example: \langle \mathbb{R}, d \rangle is a metric space on real numbers, where d(x, y) = |x - y|.

Theorem: Given a set of metric space \langle X_1, d_1 \rangle, \langle X_2, d_2 \rangle, \cdots, \langle X_i, d_i \rangle, we can obtain a metric space on X = \Pi_{i = 1}^n X_i with d: X \times X \to \mathbb{R}. Let d(x, y) = \max_{i \leq i \leq n} {d_i(x_i, y_i)}

Proof: By verifying the four requirements in the definition. □

Remark: By instantiating the above theorem, \langle \mathbb{R}^n, d: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R} \rangle is a metric space, where d((x_1, x_2, \cdots, x_n), (y_1, y_2, \cdots, y_n)) = \max_{i \leq i \leq n}{|x_i - y_i|}

Remark: \langle \mathbb{R}^n, d{}’: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R} \rangle is a metric space, where d{}’((x_1, x_2, \cdots, x_n), (y_1, y_2, \cdots, y_n)) = \sqrt{\Sigma_{i=1}^n{ (x_i - y_i) }^2} Note: d{}’ is the Euclidean distance function.

##### 1.2` `Continuity

Definition (Continuity): A function f: \mathbb{R} \to \mathbb{R} is continuous at point a \in \mathbb{R}, if given \epsilon > 0, there exists a \delta > 0, such that if |x - a| < \delta, then |f(x) - f(a)| < \epsilon for all x \in \mathbb{R}.

The function is continuous if it is continuous at each point of \mathbb{R}.

Definition (Continuity): Given two metric spaces \langle X, d \rangle and \langle Y, d{}’ \rangle, a function f: X \to Y is continuous at point a \in X, if given \epsilon > 0, there exists a \delta > 0, such that if d(x, a) < \delta, then d{}’(f(x), f(a)) < \epsilon for all x \in X.

The function is continuous if it is continuous at each point of X.

Theorem: A constant function f : X \to Y is continuous.

Proof: It is always the case that d{}’(f(x), f(a)) = 0. □

Theorem: The identity function f : X \to X is continuous.

Proof: Choose \delta = \epsilon. □

Theorem (Composition of Continuous Functions): Let \langle X, d_1 \rangle, \langle Y, d_2 \rangle, and \langle Z, d_3 \rangle be metric spaces. Let f : X \to Y be continuous at a \in X, and g: Y \to Z be continuous at f(a) \in Y, then g \circ f : X \to Z is continuous at a \in X.

Corollary: Let \langle X, d_1 \rangle, \langle Y, d_2 \rangle, and \langle Z, d_3 \rangle be metric spaces. Let f : X \to Y and g: Y \to Z be continuous, then g \circ f : X \to Z is continuous.

##### 1.3` `Open Balls

Definition (Open Ball): Let \langle X, d \rangle be a metric space, and a \in X, and \delta > 0. The subset \{ x \in X | d(a, x) < \delta \} \subseteq X is called the open ball about a of radius \delta. We denote that subset by B(a;\delta).

In other words, x \in B(a;\delta) iff x \in X and d(x, a) < \delta. Similarly, if \langle Y, d{}’ \rangle is a metric space, and f: X \to Y, then we have y \in B(f(a); \epsilon) iff y \in Y and d{}’(y, f(a)) < \epsilon.

Theorem (4.2): A function f : \langle X, d \rangle \to \langle Y, d{}’ \rangle is continuous at a point a \in X iff for all \epsilon > 0, there exists a \delta > 0 such that f(B(a; \delta)) \subset B(f(a); \epsilon)

Remark: For a function f : X \to Y, we have F(U) \subset V iff U \subset f^{-1}(V).

Theorem (4.3): A function f : \langle X, d \rangle \to \langle Y, d{}’ \rangle is continuous at a point a \in X iff for all \epsilon > 0, there exists a \delta > 0 such that B(a; \delta) \subset f^{-1}(B(f(a); \epsilon))

##### 1.4` `Neighborhoods

Definition (Neighborhoods): Let \langle X, d\rangle be a metric space and a \in X. A set N \subset X is called a neighborhood of a if there exists a \delta > 0 such that B(a; \delta) \subset N

The collection \mathcal{N}_a of all neighborhoods of a \in X is called a complete system of neighborhoods of the point a.

Lemma (4.5): Let \langle X, d\rangle be a metric space and a \in X. For every \delta > 0, the open ball B(a; \delta) is a neighborhood of point b \in B(a; \delta).

Proof: We need to show that for all b \in B(a; \delta), there exists \eta > 0 such that B(b; \eta) \subset B(a; \delta). Choose \eta < \delta - d(a, b). Then \forall x \in B(b; \eta), d(a, x) \leq d(a, b) + d(b, x) < d(a, b) + \eta < d(a, b) + \delta - d(a, b) = \delta Therefore, \forall x \in B(b; \eta), we have x \in B(a; \delta). □

Lemma (4.6): Let f:\langle X, d\rangle\to\langle Y, d{}’\rangle. f is continuous at point a \in X iff for for all neighborhood M of f(a), there exists a corresponding neighborhood N of a, such that f(N) \subset M \text{, or equivalently } N \subset f^{-1}(M)

Proof:

First assume that f is continuous at a. Let M be a neighborhood of f(a). We must to show that there exists a neighborhood N of a and f(N) \subset M. 1) By the definition of neighborhoods, there exists \epsilon such that B(f(a), \epsilon) \subset M. 2) Since f is continuous, there exists a \delta > 0 such that f(B(a; \delta)) \subset B(f(a); \epsilon) (theorem 4.2). 3) Now, let N be B(a; \delta). We can conclude that f(N) = f(B(a; \delta) \subset B(f(a); \epsilon) \subset M

To prove the other direction, assume that \forall neighborhood M of f(a), there exists a neighborhood N of a, such that f(N) \subset M. Let \epsilon > 0 be the threshold of such neighborhood M = B(f(a); \epsilon). To prove f is continuous, we must show that there exists \delta > 0 such that f(B(a; \delta)) \subset B(f(a); \epsilon) (theorem 4.2). By the hypothesis, there is neighborhood N of a such that f(N) \subset M. By the definition of neighborhood, there exists \delta such that B(a; \delta) \subset N. Therefore, we have f(B(a; \delta)) \subset f(N) \subset M = B(f(a); \epsilon) □

Lemma (4.7): Let f:\langle X, d\rangle\to\langle Y, d{}’\rangle. f is continuous at point a \in X iff for each neighborhood M of f(a), f^{-1}(M) is a neighborhood of a.

Theorem (4.8): Let \langle X, d\rangle be a metric space.

\forall a \in X, there exists at least one neighborhood of a.

\forall a \in X and its neighborhood N, a \in N.

\forall a \in X, if N is its neighborhood and N{}’ \supset N, then N{}’ is also a neighborhood of a.

\forall a \in X and its neighborhoods N and M, N \cap M is also a neighborhood of a.

\forall a \in X and each neighborhood N of a, there exists a neighborhood O of a, such that O \subset N and O is a neighborhood of points in O.

Definition (Basis of Neighborhoods): Let a be a point in metric space X. A collection \mathcal{B}_a of neighborhoods of a is called a basic for the neighborhood system at a, if every neighborhood N of a contains some element B of \mathcal{B}_a.

##### 1.5` `Limits

First, recall the definition of limit on real line.

Definition (Limit): a \in \mathbb{R} is the limit of the sequence a_1, a_2, \cdots if given \epsilon > 0, there exists a positive integer N such that, for all n > N, we have |a - a_n| < \epsilon. We can also say the sequence converges to a, and write \lim_n a_n = a.

Definition (Limit, generalized): Let \langle X, d\rangle be a metric space. Let a_1, a_2, \cdots be a sequence of points in X. A point a \in X is the limit of the sequence if \lim_n d(a, a_n) = 0

Corollary (5.3): Let \langle X, d\rangle be a metric space and a_1, a_2, \cdots be a sequence of points in X. \exists a \in X, \lim_n a_n = a \Leftrightarrow for each neighborhood V of a, \exists N \in \mathbb{N}, \forall n > N, a_n \in V.

Proof:

Let V be a neighborhood of a. By definition of neighborhoods, there is \epsilon such that a \in B(a; \epsilon) \subset V. Then if \lim_{n} a_n = a, there exists an ineger N, such that d(a, a_n) < \epsilon whenever n > N. Therefore a_n \in V.

Given \epsilon > 0 and B(a; \epsilon) is a neighborhood of a. By premises, we have N such that n > N, a_n \in B(a; \epsilon). Then d(a, a_n) < \epsilon, and therefore \lim_{n} a_n = a. □

If S is a set of infinite points, and given a proposition P, there is a finite subset X \subset S, such that \forall x \in X, P(x), then we can say the P is true for almost all the elements in S.

Therefore, \lim_{n} a_n = a if for each neighborhood V of a, almost all points a_n are in V. In other wors, for each V, there are at most finite elements in sequence a_n are not in V.

Theorem (5.4): Let \langle X, d\rangle and \langle Y, d{}’\rangle be two metric space. A function f : X \to Y is continuous at a point a \in X if and only if, if \lim_n a_n = a for a sequence of points in X, \lim_n f(a_n) = f(a).

Proof: TODO.

If \lim_n a_n = a, then by a substitution, \lim_n f(a_n) = f(a) is equivalent to \lim_n f(a_n) = f(\lim_n a_n). In other words, the continuous function commutes with the limit operation.

Definition (bounds): Let X be a set of real numbers. A number b is an upper bound of A if \forall x \in A, x \leq b. A number c is a lower bound of A if \forall x \in A, c \leq x. If A has both an upper bound and a lower bound, then A is said to be bounded. An upper bound b^* is a least upper bound of A, if for all upper bound b of A, b^* \leq b. A lower bound c^* is a greatest lower bound of A, if for all lower bound c of A, c \leq c^*.

Definition (Completeness postulate): Completeness postulateis a properties of the real number systems, stating that if a non-empty set A of real numbers has an upper bound, then it also has a lub; and if a non-empty set B of real numbers has a lower bound, then it also has a \text{glb} .

Lemma (5.6): Let b be a \text{glb} of a non-empty set A. For each \epsilon > 0, exists an element x \in A such that x - b < \epsilon.

Proof: By contradiction. □

Corollary (5.7): Let b be a \text{glb} of a non-empty set A of real numbers. There is a sequence a_n of real numbers such that a_n \in A and \lim_n a_n = b.

Definition (Distance between an element and a set): Let \langle X, d\rangle be a metric space. Let a \in X and A be a non-empty subset of X. The \text{glb} of \{ d(a, x) | x \in A \} is called the distance between a and A and is denoted by d(a, A).

##### 1.6` `Open Sets and Closed Sets

##### 1.7` `Subspaces

##### 1.8` `Equivalence of Metric Spaces

#### 2` `Topological Space

History

03/28/2020 - updated until sec. 1.3

03/16/2020 - added the post