Notes on Topology🔗

This post is the note from reading Bert Mendelson’s book Introduction to Topology, 3rd Edition. Most of the content of this note directly comes from the book. The first chapter of the book about the set theory is omitted in this note.

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 $d(x, y) = |x - y|$.

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 (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 open ball $B(a;\delta)$ about $a$ of radius $\delta$ is $\{ x \in X | d(a, x) < \delta \} \subseteq X$.

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 (4.7): 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 $$\lim_n d(a, a_n) = 0$$

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 $V$.

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 $\text{glb}$ .

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🔗

Definition (Open Set): A subset $O$ of a metric space is open if $O$ is a neighborhood of each of its points.

We shall also use "open balls" to defined "open sets".

Theorem (6.4): Let $\langle X, d\rangle$ be a metric space.

• The empty set is open

• $X$ is open

• If $O_1, O_2, \cdots, O_n$ are open, $O_1 \cap O_2 \cdots \cap O_n$ is also open.

• If $O_1, O_2, \cdots, O_n$ are open, $O_1 \cup O_2 \cdots \cup O_n$ is also open.

Definition (Closedness): A subset $F$ of a metric space is close if its complement $C(F)$ is open.

A set can be both open and closed. For example, given a metric space $\langle X, d\rangle$, $\emptyset$ and $X$ are open, therefore their complement $X$ and $\emptyset$ are closed. So, $\emptyset$ and $X$ are both open and closed.

Definition (Limit Point): Let $A$ be a subset of metric space $X$. A point $b \in X$ is a limit point of $A$ if every neighborhood of $b$ contains a point of $A$ different from $b$.

1.7 Subspaces of Metric Spaces🔗

1.8 Equivalence of Metric Spaces🔗

2 Topological Spaces🔗

2.1 Definition of Topological Spaces🔗

2.2 Neighborhood Spaces🔗

2.3 Closure, Interior, and Boundary🔗

2.4 Continuity and Homeomorphism🔗

2.5 Subspaces of Topological Spaces🔗

2.6 Products🔗

2.7 Indentification Topologies🔗

2.8 Categories and Functors🔗

3 Connectedness🔗

4 Compactness🔗

History

03/28/2020 - updated until sec. 1.3

03/16/2020 - added the post