# 0x001 Topology This note mainly deals with topological space and metric space, normed space and inner product spaces will go to the algebra and analysis pages.

## 1. Topological Space

Definition (topology) Let $$X$$ be a non-empty set. A set $$\mathcal{T}$$ of subsets of $$X$$ is said to be topology on $$X$$ iff

• $$X \in \mathcal{T}, \emptyset \in \mathcal{T}$$
• union of any number of sets of $$\mathcal{T}$$ belongs to $$\mathcal{T}$$
• interaction of any two sets of $$\mathcal{T}$$ belongs to $$\mathcal{T}$$

The pair $$(X, \mathcal{T})$$ are called topological space

discrete, indiscrete topology

There are two trivial topological space for any $$X$$, $$2^X$$ is the discrete topology of $$X$$, $$\{ \emptyset, X \}$$ is the indiscrete topology of $$X$$.

finite complement topology

Let $$X$$ be a set, $$\mathcal{T}$$ be the collection of all subsets $$U \subset X$$ such that $$X - U$$ is either finite or all of $$X$$

Definition (finer, coarser) Suppose $$\mathcal{T}, \mathcal{T'}$$ are two different topologies on a given set $$X$$, if $$\mathcal{T} \subset \mathcal{T'}$$, then $$\mathcal{T}$$ is a coarser and $$\mathcal{T'}$$ is a finer topology

Definition (open set) $$U \subset X$$ is an open set iff $$U \in \mathcal{T}$$

### 1.1. Basis of Topology

Defining the entire set of $$\mathcal{T}$$ is too difficult, we can instead specify a smaller collection of subset of $$X$$ and define topology on top of that. This is called basis.

Definition (basis) If $$X$$ is a set, a basis of a topology on $$X$$ is a collection $$\mathcal{B}$$ of subsets such that

• for $$x \in X$$, there exists a $$B \in \mathcal{B}$$ containing it
• if $$x \in B_1 \cap B_2$$, then there exists a $$B_3 \subset B_1 \cap B_2$$ that contains $$x$$

If $$\mathcal{B}$$ satisfies these conditions, the topology $$\mathcal{T}$$ defined by $$\mathcal{B}$$

Definition (open set, topology generated by a basis) A subset $$U$$ of $$X$$ is called open if for every $$x \in U$$, there exists a basis element $$B$$ such that $$x \in B$$ and $$B \subset U$$

For example, the collection of one-point set of $$X$$ is the basis of the discrete topology of $$X$$

By using the basis definition, we can characterize each open set $$U$$ can be expressed as a union of basis elements.

We can also use topology to generate a basis

Lemma (basis generated by a topology) Let $$X$$ be a topological space. Suppose $$\mathcal{C}$$ is a collection of open set such that for each open set $$U$$ in $$X$$ and every point $$x \in U$$, there exists an element $$C \in \mathcal{C}$$ such that $$x \in C \subset U$$. Then $$\mathcal{C}$$ is a basis for the topology of $$X$$

Definition (subbasis) A subbasis $$\mathcal{S}$$ for a topology $$X$$ is a collection of subsets of $$X$$ whose union equals $$X$$. The topology generated by $$\mathcal{S}$$ is defined to be the collection of all unions of all finite intersection of $$\mathcal{S}$$

Subbasis $$\mathcal{S}$$ first creates a basis $$\mathcal{B}$$ by taking the finite intersection, then the basis generates the topology by taking union

Definition (order topology) Let $$X$$ be a set with simple order relation. Assume $$X$$ has more than one element, then the following collection $$\mathcal{B}$$ is a basis for a topology on $$X$$, called order topology

• All open set $$(a,b)$$
• All interval $$[-\infty, b)$$ if the smallest $$-\infty$$ exists
• All interval $$(a, \infty]$$ if the largest $$\infty$$ exists

### 1.2. Product Topology

Definition (product topology) Let $$X,Y$$ be topological spaces. The product topology on $$X \times Y$$ is the topology having as the basis $$\mathcal{B}$$ of all sets of the form $$U \times V$$ where $$U,V$$ are open subset of $$X,Y$$

### 1.3. Quotient Topology

Definition (quotient map) Let $$X,Y$$ be topological spaces, let $$p X \to Y$$ be a surjective map, the map is said to be a quotient map if a subset $$U$$ is open in $$Y$$ iff $$p^{-1}(U)$$ is open in $$X$$.

Definition (quotient topology) If $$X$$ is a space and $$A$$ is a set and if $$p: X \to A$$ is a surjective map, then there exists exactly one topology $$\Tau$$ on $$A$$ relative to which $$p$$ is a quotient map. The topology is defined such that $$U$$ is open if $$p^{-1}(U)$$ is open.

Definition (quotient space) Let $$X$$ be a topological sapce, $$X^*$$ be a partition of $$X$$ into disjoint subsets whose union is $$X$$. Let $$p: X \to X^*$$ be a surjective map that carries each point of $$X$$ to the elements of $$X^*$$ containing it. In the quotient topology induced by $$p$$, the space $$X^*$$ is called a quotient space of $$X$$.

### 1.4. Metric Topology

Definition (metric) A metric on a set $$X$$ is a function:

$d: X \times X \to R$

having the following properties:

• $$d(x,y) \geq 0$$
• $$d(x,y) = d(y,x)$$
• $$d(x,y) + d(y,z) \geq d(x,z)$$

Definition (metric topology) If $$d$$ is a metric on $$X$$, the collection of all $$\epsilon$$-balls $$B_d(x, \epsilon)$$ for $$x \in X, \epsilon > 0$$ is a basis for a topology of $$X$$, called the metric topology induced by $$d$$

Definition (metrizable space) If $$X$$ is a topological space, $$X$$ is said to be metrizable if there exists a metric $$d$$ on the set $$X$$ that induces the topology of $$X$$

Metrizability is always a desirable property for a space, therefore a important problem is to find conditions on a topological space that will guarantee it is metrizable. For example, Urysohn’s metrization theorem.

### 1.5. Subspace Topology

Definition (subspace) Let $$X$$ be a topological space with topology $$\mathcal{T}$$. If $$Y$$ is a subset of $$X$$, the collection

$\mathcal{T}_Y = \{ Y \cap U | U \in \mathcal{T} \}$

is a topology on $$Y$$, called the subspace topology

Definition (subspace basis) If $$\mathcal{B}$$ is a basis for the topology of $$X$$ then the collection

$\mathcal{B}_Y = \{ B \cap Y | B \in \mathcal{B} \}$

is a basis of the subspace topology on $$Y$$

### 1.6. Closed Set and Limit Points

Definition (neighborhood) $$U$$ is a neighborhood of $$x$$ if $$U$$ is a open set containing $$x$$

Note there are some book defining neighborhood to be a subset contains an open set containing $$x$$: neighborhood does not need to be open.

Definition (closed) A subset of $$A$$ of a topological space $$X$$ is closed if the set $$X - A$$ is open

Definition (interior, closure)

• the interior of $$A$$ is defined as the union of all open sets contained in $$A$$
• the closure of $$A$$ is defined to be the intersection of all closed sets containing $$A$$

Lemma (closure) if $$x \in \bar{A}$$ then every neighborhood of $$x$$ intersect $$A$$

Lemma (closure in subspace) Let $$Y$$ be a subspace of $$X$$, $$A$$ be a subset of $$Y$$. let $$\bar{A}$$ be the closure of $$A$$ in $$X$$. Then the closure of $$A$$ in $$Y$$ is $$\bar{A} \cap Y$$

Definition (limit point) We say $$x$$ is a limit point of $$A$$ if every neighborhood of $$x$$ intersect $$A$$ in some point other than $$x$$ itself.

Definition (convergence) A sequence $$x_1, ...$$ of points in $$X$$ converges to $$x$$ if for each neighborhood $$U$$ of $$x$$, there exists an $$N$$ such that all $$n \geq N$$, $$x_n \in U$$.

convergence to two points

Consider the topology $$\{ \emptyset, \{a, b\}, \{b, c\}, \{b\}, \{a,b,c \} \}$$ on $$\{a, b, c \}$$. The sequence $$b, b, b...$$ converges to $$a,b,c$$

Definition (Fréchet, T1) For every pair of distinct point, each has a neighborhood that does not contain the other

The Hausdorff condition is generally considered to be a very mild extra condition to impose on a topological space.

Definition (Hausdorff, T2) A topological space $$X$$ is called a Hausdorff space if for every distinct point $$x_1, x_2$$, there exists disjoint open neighborhoods $$U_1, U_2$$ such that $$x_1 \in U_1, y_1 \in U_2$$

Theorem (unique convergence) If $$X$$ is Hausdorff, then a sequence of points on $$X$$ converges to at most one point of $$X$$

## 2. Continuous Function

Continuity is an definition not only depending on function itself, but also on topologies of $$X,Y$$

Definition (continuity) Let $$X,Y$$ be a topological spaces, a function $$f: X \to Y$$ is said to be continuous if for each open subset $$V$$ of $$Y$$, the inverse image $$f^{-1}(V)$$ is an open set of $$X$$

Condition (continuity) To prove continuity of $$f$$, it is sufficient to check the inverse of basis or subbasis element:

• the inverse of every basis element of $$f$$ is open
• the inverse of every subbasis element is open

Condition (continuity) The followings are equivalent

• $$f$$ is continuous
• for every subset $$A$$, $$f(\bar{A}) \subset \overline{f(A)}$$
• For every closed subset $$B \subset Y$$, the set $$f^{-1}(B)$$ is also closed in $$X$$
• For each $$x \in X$$ and each neighborhood $$V$$ of $$f(x)$$, there is a neighborhood $$U$$ of $$x$$ such that $$f(U) \subset V$$

### 2.1. Homeomorphism

Homeomorphism is a bijective correspondence that preserves topological properties (properties defined using open sets). It is analogous to isomorphism which preserves algebraic structure.

Definition (homeomorphism) Let $$X,Y$$ be topological spaces, let $$f: X \to Y$$ be bijection. If both $$f$$ and its inverse are continuous, then $$f$$ is called homeomorphism

Lemma (homeomorphism) another way to say homeomorphism is that $$f$$ is a bijective such that $$f(U)$$ is open iff $$U$$ is open.

One popular example of homeomorphism is the Poincaré conjecture

statement of Poincaré conjecture

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

Definition (topological imbedding) Suppose $$f: X \to Y$$ is an injective continuous mapping, let $$Z$$ be the image set of $$f(X)$$ and suppose the restriction $$f': X \to Z$$ is a homeomorphism then we say the map $$f: X \to Y$$ is a topological imbedding of $$X$$ \in $$Y$$

### 2.2. Construction

Theorem (construction of continuous functions) Let $$X,Y,Z$$ be topological spaces

• (constant function) If $$f: X \to Y$$ maps all of $$X$$ into the single point $$y_0$$ of $$Y$$. Then $$f$$ is continuous
• (inclusion) If $$A$$ is a subspace of $$X$$, the inclusion function $$j: A \to X$$ is continuous
• (composite) If $$f: X \to Y, g: Y \to Z$$ are continous, then $$g \circ f: X \to Z$$ is continuous
• (domain restriction) If $$f: X \to Y$$ is continuous, if $$A$$ is a subspace of $$X$$, then restricted functio n$$f|A: A \to Y$$ is continuous
• (range restriction or expansion) Let $$f: X \to Y$$ be continuous, If $$Z$$ is a subspace of $$Y$$ containing the image set $$f(X)$$ or $$Z$$ is a space containing $$Y$$ as subspace, the function $$f: X \to Z$$ is continuous
• (local formulation of continuity) The map $$f: X \to Y$$ is continuous if $$X$$ can be written as a union of $$U_{\alpha}$$ such that $$f|_{U_{\alpha}}$$ is continuous for every $$\alpha$$

Theorem (maps into products) Let $$A \to X \times Y$$ be the given equation

$f(a) = (f_1(a), f_2(a))$

Then $$f$$ is continuous iff the function

$f_1: A \to X, f_2: A \to Y$

are continuous

## 3. Connectedness and Compactness

### 3.1. Connectedness

The concept of connectedness is useful to prove the intermediate value theorem. It is a topological property as it is formulated entirely in terms of collection of open sets

Definition (connected space) Let $$X$$ be a topological space, A separation of $$X$$ is a pair $$U,V$$ of disjoint open subsets of $$X$$ whose union is $$X$$. The space is said to be connected if there is no separation of $$X$$.

Another way to state connectedness: A space $$X$$ is connected iff the only subsets of $$X$$ that are both open and closed are the empty set and $$X$$ itself

Lemma (connected subspace) If $$Y$$ is a subspace of $$X$$, a separation of $$Y$$ if a pair of disjoint sets $$A,B$$ where $$Y=A \cap B$$, and $$\bar{A} \cap B$$ and $$A \cap \bar{B}$$ are both empty. The space $$Y$$ is connected if there exists no separation of $$Y$$

Roughly $$\bar{A} \cap B \neq \emptyset$$ means $$A$$ has a limit point in $$B$$ therefore $$A$$ is not closed and $$B$$ is not open.

rational is not connected

$$Q$$ is not connected. We can write Q as $$A \cup B$$ where

$A = Q \cap (-\infty, \sqrt{2}), B = Q \cap (\sqrt{2}, +\infty)$

where $$\hat{A} \cap B = A \cap \hat{B} = \emptyset$$

Lemma A set $$E \subset R$$ is connected iff whenever $$a < c < b$$ with $$a,b\in E$$ it follows that $$c \in E$$

The following theorem generalizes the intermediate value theorem

Theorem (continuity preserves connectedness) The image of a connected subspace under a continuous map is connected.

### 3.2. Compactness

compactness is a concept generalizing a bounded closed set.

Definition (open cover) Suppose $$A \subset R$$, a collection $$\mathcal{C}$$ of open subsets of $$R$$ is called an open cover of $$A$$ if $$A$$ is contained in the union of all the sets in $$\mathcal{C}$$

Definition (finite subcover) An open cover $$\mathcal{C}$$ of $$A$$ is said to have a finite subcover if $$A$$ is contained in the union of some finite list of sets in $$\mathcal{C}$$

Definition (sequential compactness) A set $$K \subset R$$ is sequential-compact if every sequence in $$K$$ has a subsequence that converges to some limit in $$K$$.

Definition (compactness) A set $$K \subset R$$ is compact if every open cover for $$K$$ has a finite subcover

For example, if the set $$K$$ only contains finite elements, then obviously it is compact.

realline is not compact

The real line $$R$$ is not compact. Consider a cover of

$\{ (n, n+2) | n \in Z \}$

It does not have any finite subcover

compact example

The following set is compact

$A = \{ 0 \} + \{ 1/n | n \in Z^{+} \}$

For any open cover of $$A$$, we can pick an open set $$U$$ containing $$A$$, this set $$U$$ should contain infinitely many tails $$1/n$$, the remaining elements are finite, so it is easy to pick up their cover.

Theorem (Heine-Borel) Following statements are equivalent in $$\mathbb{R}$$

1. $$K$$ is sequential compact
2. $$K$$ is closed and bounded
3. $$K$$ is compact

Proof

• $$1 \to 2$$: Suppose $$K$$ is not bounded, then we can pick up a sequence $$(x)_n$$ such that $$x_n \geq n$$, which suggests any subsequence $$x_{n_k} \geq n_k$$ is also not bounded. However, compactness can select a convergent subsequence, and this subsequence must be bounded. therefore, we have a contradiction. Next, consider any limit point $$x$$ for $$K$$, there must be a sequence $$(x)_n \in K$$ converging to $$x$$, then any subsuquence must converge to $$x$$ including the sequence picked up by compactness, the compactness says the convergent point must be in $$K$$ so we know $$x \in K$$, so all limit point is contained and $$K$$ is closed

• $$3 \to 2$$: To show the boundedness, construct a cover consisting of open balls with radius 1 at every $$x \in K$$, then compactness gives a finite subcover, then take the max and min. To show the closedness, suppose it is not closed: there exists a limit point $$y \notin K$$ and $$y_n \in K \to y$$. For every point $$x \in K$$, construct a open ball with radius $$|x-y|/2$$, compactness again gives a finite subcover where we can take the max, the max however is strictly less than $$y$$, therefore the finite subcover cannot cover tails of $$y_n$$

• $$2 \to 3$$: This is actually the Heine Borel Theorem. The basic idea is to first show $$[a,b]$$ is compact by consider the sup of finite covered element, and use this fact to show a random closed bounded set is compact.

### 3.3. Limit Point Compactness

Definition (limit point compactness) A space $$X$$ is said to be limit point compact if every infinite subset of $$X$$ has a limit point

Definition (sequentially compact) Let $$X$$ be a topological space and $$x_n$$ be a sequence of $$X$$. The space $$X$$ is said to be sequentially compact if every sequence of $$X$$ has a convergence subsequence

Theorem (compactness equivalence over metrizable space) Let $$X$$ be a metrizable space. Then the following are equivalent:

• compact
• limit point compact
• sequential compact

### 3.4. Local Compactness

Definition (local compactness) A space $$X$$ is said to be locally compactness at $$x$$ if there is some compact subspace $$C$$ of $$X$$ contains a neighborhood of $$x$$. If $$X$$ is locally compact at every point $$x$$, then $$X$$ is said to be local compact.

## 5. Baire Spaces

Definition (Baire space) A space is said to be a Baire space if given any countable collection $$\{ A_n \}$$ of closed sets of $$X$$ each of which has empty interior in $$X$$, their union $$\cup A_n$$ also has empty interior in $$X$$.

Definition (Baire category theorem) If $$X$$ is a compact Hausdorff space or a complete metric space, then $$X$$ is a Baire space

## 6. Fundamental Group

One of the basic problem of topology is to determine whether two given topological space are homeomorphic or not.

### 6.1. Homotopy of Paths

Definition (homotopy) If $$f, f'$$ are continuous map of spaces $$X \to Y$$. We say $$f$$ is homotopic to $$f'$$ if there is a continuous map $$F: X \times I \to Y$$ such that for each $$x$$

$F(x, 0) = f, F(x, 1) = f'$

The map $$F$$ is called a homotopy between $$f$$ and $$f'$$. If $$f$$ is homotopic to $$f'$$, we write $$f \simeq f'$$

## 7. Reference

 Topology 2nd Edition Munkres

 Abbott, Stephen. Understanding analysis. Vol. 2. New York: Springer, 2001.

 Tao, Terrence. "Analysis (Volume 1)." Hindustan Book Agency (2006).