Overview §

Columbia University, Fall 2024 – R. Akhmechet

Course description

Metric spaces, continuity, compactness, quotient spaces. The fundamental group of topological space. Examples from knot theory and surfaces. Covering spaces.

Study status §

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Topics §

Topological spaces and continuous functions §

Topological spaces and open sets §

Definition: Topology, topological space

A topology on a set $X$ is a collection $\mathcal{T}$ of subsets of $X$ (i.e., a subset of the power set $\mathcal{T}\subseteq \mathcal{P}(X)$) that has the following properties:

1. $\varnothing$ and $X$ are in $\mathcal{T}$;
2. The union of the elements of any (arbitrary) sub-collection of $\mathcal{T}$ is also in $\mathcal{T}$;
3. The intersection of the elements of any finite sub-collection of $\mathcal{T}$ is also in $\mathcal{T}$.

A topological space is an ordered pair $(X,\mathcal{T})$, where $X$ is a set and $\mathcal{T}$ is a topology on $X$.

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Definition: Finer, coarser, and comparable topologies

Two topologies $\mathcal{T},\mathcal{T}’$ on a set $X$ are comparable if either $\mathcal{T}\subseteq \mathcal{T}’$ or $\mathcal{T}’\subseteq \mathcal{T}$. We say $\mathcal{T}’$ is finer than $\mathcal{T}$ if $\mathcal{T}’$ has more open sets than $\mathcal{T}$; that is, if $\mathcal{T}\subseteq \mathcal{T}’$. In this case, we also say $\mathcal{T}$ is coarser than $\mathcal{T}’$.

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Subspace (induced) topology §

Definition: Subspace (induced) topology

Given a topological space $(X,\mathcal{T})$ and a subset $A\subseteq X$, the subspace or induced topology on $A$ is defined by the collection

$${\mathcal{T}}_A=\{A\cap U|U\in \mathcal{T}\}.$$

In other words, open sets in $A$ (i.e., a subset of $A$ that is open in the subset topology) are obtained by intersecting $A$ with open sets of $X$ (i.e., an open set in $(X,\mathcal{T}$)).

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Topological bases and subbases §

Definition: Basis for a topology

Given a set $X$, a basis for a topology on $X$ is a collection $\mathcal{B}\subseteq \mathcal{P}(X)$ of subsets of $X$ such that:

• (B1) For each $x\in X$, there exists at least one basis element $B$ containing $x$ (that is, $\mathcal{B}$ covers $X$);
• (B2) Given two basis elements $B_1,B_2$ and a point in their intersection $x\in B_1\cap B_2$, there exists a basis element $B_3$ such that $x\in B_3$ and $B_3\subseteq B_1\cap B_2$.

Equivalently, condition (B1) says that ${\bigcup }_{B\in \mathcal{B}}B=X$, while (B2) says that if $B_1,B_2\in \mathcal{B}$, then $B_1\cap B_2$ is a union of elements (sets) in $\mathcal{B}$.

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Definition: Topology generated by a basis

Given a collection $\mathcal{B}$ which satisfies the two conditions of a basis, a subset $U\subset X$ is said to be open in $X$ if for all $x\in U$, there exists a basis element $B\in \mathcal{B}$ such that $x\in B$ and $B\subset U$.

The topology $\mathcal{T}$ generated by $\mathcal{B}$ is defined as the collection of all subsets $U$ that are open in $X$:

$$\mathcal{T}=\{U\subset X|\forall x\in U\exists B\in \mathcal{B}\text{ such that }x\in B\subset U\}.$$

Equivalently, $U\in \mathcal{T}$ if and only if $U$ is a union of basis elements $B\in \mathcal{B}$ (see Munkres, Lemma 13.1).

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Lemma (Munkres 13.2): Obtaining a basis for a given topology

Let $(X,\mathcal{T})$ be a topological space and let $\mathcal{C}\subseteq \mathcal{T}$ be a collection of open sets of $X$ such that for each open set $U\subseteq X$ and each $x\in U$, there exists $C\in \mathcal{C}$ such that $x\in C\subseteq U$. Then (1) $\mathcal{C}$ is a basis and (2) $\mathcal{C}$ generates $\mathcal{T}$.

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Lemma: Basis for a subspace topology

If $\mathcal{B}$ is a basis for the topology $\mathcal{T}$ of $X$ and $A\subseteq X$ is any subset, then

$${\mathcal{B}}^{\prime }=\{A\cap U|U\in \mathcal{B}\}$$

is a basis for the subspace topology on $A$.

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Lemma (Munkres 13.3): Using bases to determine which topology is finer

Let $\mathcal{B},\mathcal{B}’$ be bases for topologies $\mathcal{T},\mathcal{T}’$ on $X$. The following are equivalent:

1. $\mathcal{T}’$ is finer than $\mathcal{T}$ (i.e., $\mathcal{T}\subseteq \mathcal{T}’$);
2. For all $U\in \mathcal{B}$ and all $x\in U$, there exists $V\in \mathcal{B}’$ such that $x\in V\subseteq U$.

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The product topology on the Cartesian product §

Definition: Product topology on the Cartesian product X \times Y

Given topological spaces $(X,{\mathcal{T}}_X)$ and $(Y,{\mathcal{T}}_Y)$, the product topology on their Cartesian product $X\times Y$ is generated by the basis

$$\mathcal{B}=\{U\times V|U\in {\mathcal{T}}_X,V\in {\mathcal{T}}_Y\}.$$

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Closed sets and closures in topological spaces §

Definition: Closed set in topological space

Given a topological space $X$, we say a subset $A\subseteq X$ is closed if its complement $X\backslash A$ is open.

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Definition: Closure and interior of a set

Given a subset $A$ of a topological space $X$:

• The interior of $A$, denoted $\text{Int}(A)$, is the union of all open sets contained in $A$;
• The closure of $A$, denoted $\bar{A}$, is the intersection of all closed sets containing $A$.

By definition, $\text{Int}(A)\subset A\subset \bar{A}$. Further, $\text{Int}(A)$ is the biggest open set contained in $A$, and $\bar{A}$ is the smallest closed set containing $A$.

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Theorem (Munkres 17.1): Doing topology with closed sets

Let $X$ be a topological space. Then:

1. $\varnothing$ and $X$ are closed;
2. Arbitrary intersections of closed sets are closed;
3. Finite unions of closed sets are closed.

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Theorem (Munkres 17.2): Closed sets in a subspace topology

Let $Y\subset X$ be a subspace. Then a set $A\subset Y$ is closed in $Y$ if and only if it is the intersection of $Y$ with a closed set in $X$.

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Theorem (Munkres 17.4): Closure of a subset in a subspace

Let $Y\subseteq X$ be a subspace, and let $A\subset Y$ be any subset. Then the closure of $A$ in $Y$ is equal to $\bar{A}\cap Y$, where $\bar{A}$ is the closure of $A$ in $X$.

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Limit points in topological spaces §

Definition: Limit point in topological space

Given a subset $A$ of a topological space $X$, a point $x\in X$ is a limit point if every neighborhood of $x$ intersects $A$ at some point other than $x$ itself; that is, for all neighborhoods $U\subseteq X$ with $x\in U$, we have

$$A\cap (U\backslash \{x\})\ne \varnothing .$$

Equivalently, $x$ is a limit point if it is in the closure of $A\backslash \{x\}$. We write $A’$ to denote the set of all limit points of $A$ in $X$.

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Hausdorff spaces §

A sequence $x_1,x_2,\dots$ of points in an arbitrary topological space $X$ converges to the point $x$ if, for each neighborhood $U$ of $x$, there exists a positive integer $N$ such that $x_n\in U$ for all $n\ge N$.

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Definition: Hausdorff space

A topological space $X$ is called a Hausdorff space if for each pair of distinct points $x_1,x_2\in X$, there exist neighborhoods $U_1,U_2$ of $x_1,x_2$, respectively, such that $U_1\cap U_2=\varnothing$.

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Definition: The T_1 axiom

The $T_1$ axiom is the condition that all finite subsets of a space are closed.

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Theorem (Munkres 17.8 & 17.10)

If $X$ is Hausdorff, then:

• (a) Every finite subset of $X$ is closed;
• (b) Every sequence in $X$ converges to at most one point of $X$.

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Proposition (Munkres 17.9)

Let $X$ be a topological space that satisfies the $T_1$ axiom (i.e., all of its finite subsets are closed) and $A\subset X$ be any subset. Then a point $x\in A$ is a limit point of $A$ if and only if every neighborhood of $x$ contains infinitely many points of $A$.

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Continuous functions §

Definition: Continuous function in topological space

Let $X,Y$ be topological spaces. A function $f:X\to Y$ is continuous if for every open subset $V\subset Y$, the preimage set $f^{-1}(V)$ is an open subset of $X$.

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Theorem (Munkres 18.1): Equivalent statements of continuity

Let $f:X\to Y$ be a function between topological spaces $X,Y$. Then the following are equivalent:

• (a) $f$ is continuous.
• (b) For every closed set $B\subset Y$, the set $f^{-1}(B)$ is closed in $X$.
• (c) For all $x\in X$ and every neighborhood $V$ of (i.e., open set $V$ containing) $f(x)$, there is a neighborhood $U$ of $x$ such that $f(U)\subseteq V$.
• (d) For every subset $A\subset X$, we have $f(\overline{A})\subset \overline{f(A)}$, where the overline denotes the closure of a set.

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Theorem: A function is continuous if and only if its coordinate functions are continuous

Let $X\times Y$ be a product space, and ${\pi }_1:X\times Y\to X$ defined by ${\pi }_1(x,y)=x$ and ${\pi }_2:X\times Y\to Y$ defined by ${\pi }_2(x,y)=y$ be the projection maps. A function into the product space $f:Z\to X\times Y$ is uniquely determined by the coordinate functions

$$f_1={\pi }_1\circ f\text{and}{f}_2={\pi }_2\circ f,$$

which satisfy $f(z)=(f_1(z),f_2(z))$. Then $f$ is continuous if and only if $f_1,f_2$ are continuous.

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Lemma (Munkres 18.3): Pasting

Let $X=A\cup B$ for some $A,B\subseteq X$ that are both closed (or both open) in $X$, and let $f:A\to Y,g:B\to Y$ be continuous functions such that for all $x\in A\cap B$, we have $f(x)=g(x)$. Then $h:X\to Y$ defined by

$$h(x)=\{\begin{array}{r}f(x)\text{if }x\in A\\ g(x)\text{if }x\in B\end{array}$$

is continuous as well.

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Lemma ( Topology HW 3.6): Continuous functions preserve limits

If a sequence $\{x_n{\}}_{n\ge 1}$ in a topological space $X$ converges to some limit point $x\in X$, and $f:X\to Y$ is continuous, then $\{f(x_n){\}}_{n\ge 1}$ converges to $f(x)\in Y$.

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Homeomorphisms and topological embeddings §

Definition: Homeomorphism, homeomorphic

A function between topological spaces $f:X\to Y$ is a homeomorphism if $f$ is continuous, bijective, and its inverse $f^{-1}:X\to Y$ is continuous.

$X,Y$ are homeomorphic if there exists a homeomorphism between them, and we write $X\cong Y$. Equivalently, we have $X\cong Y$ if there exists continuous functions $f:X\to Y$, $g:Y\to X$ such that $g\circ f={\text{id}}_X$ and $f\circ g={\text{id}}_Y$.

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Definition: Topological embedding

A function between topological spaces $f:X\to Y$ is an embedding if it is continuous, injective, and the restriction of its range to the image of $f$, the function $f’:X\to f(X)$ defined by $x\mapsto f(x)$ is a homeomorphism.

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Metrics, metric spaces, and the metric topology §

Definition: Metric topology, metrizable space

If $(X,d)$ is a metric space (i.e., $d$ is a metric on the set $X$), for each $x\in X$, we define the $ϵ$-ball about $x$ as the set

$$B_d(x,ϵ)=\{y\in X|d(x,y)<ϵ.\}$$

Then the set of all open balls

$$\{B_d(x,ϵ)|x\in X,ϵ>0\}$$

is a basis for the metric topology induced by $d$. A general topological space $(X,\mathcal{T})$ is metrizable if there exists some metric on $X$ which induces $\mathcal{T}$.

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Definition: Standard bounded metric

Given any metric space $(X,d)$, the standard bounded metric corresponding to $d$ is defined as

$$\overline{d}(x,y)=\text{min}\{d(x,y),1\}.$$

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Generalized products §

Definition: Subbasis, topology generated by a subbasis

A subbasis for a topology on $X$ is a collection of subsets $\mathcal{S}\subseteq \mathcal{P}(X)$ such that $\mathcal{S}$ satisfies the (B1) for bases. In other words:

• For all $x\in X$, there exists $S\in \mathcal{S}$ such that $x\in S$.
• $\mathcal{S}$ covers $X$, so we have precisely ${\bigcup }_{S\in \mathcal{S}}S=X$.

The topology generated by the subbasis $\mathcal{S}$ is the topology generated by the basis

$${\mathcal{B}}_{\mathcal{S}}=\{V_1\cap \cdots \cap V_n|V_1,...,V_n\in \mathcal{S},n\ge 1\},$$

the collection of all finite intersections of elements in $\mathcal{S}$. That is, the topology generated by $\mathcal{S}$ is the union of all finite intersections of elements of $\mathcal{S}$. It is the coarsest topology in which every element of $\mathcal{S}$ is open.

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Terminology: Tuple notation for arbitrary indexing sets J be an indexing set. Given any set X, we define a J-tuple of elements of X as a function \mathbf x : J \to X. The function \mathbf x is often denoted by the quasi-tuple notation

Let

$$(x_{\alpha }{)}_{\alpha \in J}.$$

For each index $\alpha \in J$, the $\alpha$-coordinate of $\mathbf{x}$ is the value of the tuple at $\alpha$, and often denoted $x_{\alpha }$ rather than its technical definition $\mathbf{x}(\alpha )$.

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Definition: Generalized Cartesian product

Given an indexed family of sets $\{X_{\alpha }{\}}_{\alpha \in J}$, the (generalized) Cartesian product

$$\prod _{\alpha \in J}{X}_{\alpha }$$

is the family of all functions $\mathbf{x}:J\to {\bigcup }_{\alpha \in J}{X}_{\alpha }$ such that, for all $\alpha \in J$, we get an element $\mathbf{x}(\alpha )\in X_{\alpha }$.

In other words, the Cartesian product is the set of all $J$-tuples $(x_{\alpha }{)}_{\alpha \in J}$ of elements of the union of $\{X_{\alpha }{\}}_{\alpha \in J}$, where for all $\alpha \in J$, the $\alpha$-coordinate $x_{\alpha }$ is an element of the set $X_{\alpha }$.

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Definition: Box and product topologies

Suppose each set in the indexed family $\{X_{\alpha }{\}}_{\alpha \in J}$ is a topological space.

• The box topology on ${\prod }_{\alpha \in J}{X}_{\alpha }$ is the topology generated by the basis ${\prod }_{\alpha \in J}{U}_{\alpha }$, where each $U_{\alpha }$ is open in $X_{\alpha }$.
• The product topology on ${\prod }_{\alpha \in J}{X}_{\alpha }$ is the topology generated by the subbasis

$$\bigcup _{\alpha \in J}\{{\pi }^{-1}(U)|U\text{ open in }{X}_{\alpha }\},$$

which gives the basis

$$\left\{\prod _{\alpha \in J}{U}_{\alpha }|U_{\alpha }=X_{\alpha }\text{ for all but finitely many }\alpha \in J\right\}.$$

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Definition: Uniform metric, uniform topology

Consider the product ${\prod }_J\mathbb{R}$, where $J$ is any indexing set. As a set, ${\prod }_J\mathbb{R}$ is the set of all functions $J\to \mathbb{R}$, so we may also denote it ${\mathbb{R}}^J$. The uniform metric on ${\mathbb{R}}^J$ is defined by

$$\overline{\rho }(x,y)=\text{sup}\{\overline{d}(x_{\alpha },y_{\alpha })|\alpha \in J\},$$

where $\overline{d}(x,y)=\text{min}\{d(x,y),1\}$ is the standard bounded metric on $\mathbb{R}$ and $d(x,y)=|x-y|$ is the standard metric. It induces the uniform topology on ${\mathbb{R}}^J$.

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Theorem (Munkres 20.4): The uniform topology is “between” the box and product topologies

On the generalized product ${\mathbb{R}}^J$ with arbitrary indexing set $J$, the box topology is finer than the uniform topology, and the uniform topology is finer than the product topology. Further, $J$ is infinite, then all three are distinct.

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Theorem (Munkres 20.5): \mathbb R^\omega is metrizable

Let $\overline{d}(a,b)=\text{min}\{|a-b|,1\}$ be the standard bounded metric on $\mathbb{R}$. If $\mathbf{x},\mathbf{y}$ are two sequences in ${\mathbb{R}}^{\omega }$ (The countably infinite product of the real line with itself), define

$$D(x,y)=\text{sup}\left\{\frac{\overline{d}(x_i,y_i)}{i}\right\},$$

where $i>0$. Then (1) $D$ is a metric on ${\mathbb{R}}^{\omega }$ and (2) $D$ induces the product topology.

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Abstract Topology HW 3.5: Relationship between uniform convergence and the uniform metric

Let $X$ be a set and let $f_n:X\to \mathbb{R}$ be a sequence of functions. Let $\overline{\rho }$ be the uniform metric on the space ${\mathbb{R}}^X$ of all functions $X\to \mathbb{R}$. If $f_n:X\to \mathbb{R}$ is a sequence of functions, show that $(f_n)$ converges uniformly to the function $f:X\to \mathbb{R}$ if and only if the sequence $(f_n)$ converges to $f$ as elements of the metric space $({\mathbb{R}}^X,\overline{\rho })$.

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Continuous functions in general metrizable spaces §

Definition: Uniform convergence in metrizable spaces

Let $f_n:X\to Y$ be a sequence of functions from the set $X$ to the metric space $Y$. We say $(f_n)$ converges uniformly to $f:X\to Y$ if for all $ϵ>0$, there exists an integer $N$ such that for all $n\ge N$ and all $x\in X$, we have

$$d(f_n(x),f(x))<ϵ.$$

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Lemma (Munkres 21.1): \epsilon-\delta definition of continuity in metrizable spaces

Let $(X,d_X),(Y,d_Y)$ be metric spaces. Then a function $f:X\to Y$ is continuous if and only for all $x\in X$ and all $ϵ>0$, there exists $\delta >0$ such that

$$d_X(x,y)<\delta ⟹d_Y(f(x),f(y))<ϵ.$$

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Lemma (Munkres 21.2 & 21.3): Convergent sequence definition of continuity in metrizable spaces

1. Let $A\subseteq X$. If a sequence in $A$ converges to $x\in X$, then $x\in \overline{A}$, the closure of $A$.
2. Let $f:X\to Y$. If $f$ is continuous, then for all sequences $x_n\to x$ in $X$, we have $f(x_n)\to f(x)$ in $Y$.

The converse holds for metrizable spaces. As a corollary, The countably infinite product of the real line with itself in the box topology is not metrizable.

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Theorem (Munkres 21.6): Uniform limit

Let $f_n:X\to Y$ be a sequence of continuous functions from the topological space $X$ to the metric space $Y$. If $(f_n)$ converges uniformly to $f$, then $f$ is continuous.

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Connectedness and compactness §

Connectedness §

Definition: Separation, connected set in topological space

Given a topological space $(X,\mathcal{T})$, a separation of $X$ is a pair of open subsets $U,V\subset X$ which are non-empty, disjoint, and have $U\cup V=X$.

We say a set $X$ is connected if there does not exist any separation of $X$. Equivalently, $X$ is connected if and only if the only sets that are both closed and open are $\varnothing$ and $X$.

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Definition: Path, path-connected

Let $(X,\mathcal{T})$ be a topological space. Given $x,y\in X$, a path from $x\to y$ is a continuous map $p:[0,1]\to X$ such that $p(0)=x$ and $p(1)=y$.

We say $X$ is path-connected if for all $x,y\in X$, there exists a path from $x$ to $y$.

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Lemma: Facts about connectedness

• (i) If $Y\subseteq X$ is a connected subspace and $U,V$ form a separation of $X$, then $Y\subseteq U$ or $Y\subseteq V$.
• (ii) If $A\subseteq X$ is connected and $A\subseteq B\subseteq \overline{B}$, then $B$ is also connected.
• (iii) Conditions for the union of connected subspaces to be connected. Suppose $\{A_{\alpha }{\}}_{\alpha \in J}$ is a collection of connected subspaces of $X$ such that there exists a “special” index $\beta \in J$ such that for all other $\alpha \in J$, we have $A_{\beta }\cap A_{\alpha }\ne \varnothing$. Then ${\bigcup }_{\alpha \in J}{A}_{\alpha }$ is connected.

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Theorem: Every interval of the real numbers is connected.

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Proposition: If X is path-connected, then X is connected.

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Proposition: Continuous functions preserve (path-)connectedness

Let $f:X\to Y$ be a continuous function between topological spaces. Then:

• (i) If $X$ is connected, then so is $f(X)$.
• (ii) If $X$ is path-connected, then so is $f(X)$.

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Connected and path components §

Definition: (Connected) components

Given any points $x,y$ in a topological space $X$, define an equivalence relation by $x{\sim }_{C}y$ if there exists a connected subspace $A\subseteq X$ such that $x,y\in A$. The components of $X$ are the equivalence classes under ${\sim }_C$, and the component of a point $x\in X$ is denoted

$$C(x)=\{y\in X|y{\sim }_{C}x\}.$$

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Definition: Path components

Given any points $x,y$ in a topological space $X$, define an equivalence relation by $x{\sim }_{P}y$ if there exists a path from $x\to y$. The path components of $X$ are the equivalence classes under ${\sim }_P$, and the path component of a point $x\in X$ is denoted

$$P(x)=\bigcup _{\begin{array}{c}x\in A\subseteq X\\ A\text{ is path connected}\end{array}}A.$$

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Compactness §

Definition: Open cover, compact topological space

An open covering of a topological space $X$ is a collection of subsets $\mathcal{C}\subseteq \mathcal{P}(X)$ such that each $U\in \mathcal{C}$ is open in $X$ and ${\bigcup }_{U\in \mathcal{C}}=X$.

The space $X$ is compact if for every open cover $\mathcal{C}$ of $X$, there exists a sub-collection $\mathcal{C}’\subseteq \mathcal{C}$ such that $\mathcal{C}’$ is finite and also covers $X$.

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Lemma (Munkres 26.1):

A subspace $Y\subseteq X$ is compact if and only if every covering of $Y$ by open sets in $X$ contains a finite subcover (more precisely, every collection of open sets in $X$ whose union contains $Y$ has a finite sub-collection whose union also contains $Y$).

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Lemma (Munkres 26.2 & 26.3):

• (i) If $X$ is compact and $Y\subseteq X$ is closed, then $Y$ is compact.
• (ii) If $Y\subseteq X$ is compact and $X$ is Hausdorff, then $Y$ is closed in $X$.

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Theorem (Munkres 26.5 & 26.6): Continuous functions and compactness

• (i) If $f:X\to Y$ is continuous and $X$ is compact, then $f(X)$ is compact.
• (ii) Let $f:X\to Y$ be a continuous bijection. If $X$ is compact and $Y$ is Hausdorff, then $f$ is a homeomorphism.

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Theorem (Munkres 26.7): The product of finitely many compact spaces is compact.

If $X$ and $Y$ are compact, then their product $X\times Y$ is also compact. By induction, the product of finitely many compact spaces is compact.

This holds for arbitrarily many spaces in the product topology, but not for infinite products in the box topology.

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Theorem: Continuity is equals uniform continuity on compact spaces

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. If $f:X\to Y$ is continuous and $X$ is compact, then $f$ is uniformly continuous.

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Limit point and sequential compactness §

Definition: Limit point compact

A metric space $(X,d)$ is limit point compact if every infinite subset of $X$ has a limit point.

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Definition: Sequentially compact metric space

A metric space $(X,d)$ is called sequentially compact if every sequence has a convergent subsequence.

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Theorem (Munkres 28.1): Compactness implies limit point compactness in any topological space.

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Theorem (Munkres 28.2): Equivalent forms of compactness in metric spaces

Let $(X,d)$ be a metrizable space. Then the following are equivalent:

• (i) $X$ is compact;
• (ii) $X$ is limit point compact;
• (iii) $X$ is sequentially compact

Note that (i) $⟹$ (ii) in any topological space.

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Uniform continuity §

Definition: Distance between a set and a point in a metric space

Let $(X,d)$ be a metric space. If $A\subseteq X$ is nonempty and $x\in X$, the distance from $x$ to $A$ is

$$d(x,A)=\inf \{d(x,a)|a\in A\}.$$

Further, the map from $X\to \mathbb{R}$ defined by $x\mapsto d(x,A)$ for all $x\in X$ is continuous.

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Definition: Diameter of a bounded set in a metric space

Let $(X,d)$ be a metric space and $A\subseteq X$ be bounded. Then the diameter of $A$ is defined by

$$\text{diam}(A)=\text{sup}\{d(x,y)|x,y\in A\}.$$

Note that $A$ is bounded if and only if the diameter is finite.

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Theorem: Continuity is equals uniform continuity on compact spaces

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. If $f:X\to Y$ is continuous and $X$ is compact, then $f$ is uniformly continuous.

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Local compactness and compactification §

Definition: Local compactness

A topological space $X$ is locally compact at a point $x\in X$ if there exists a compact subspace $C\subseteq X$ and a neighborhood $U$ of $x$ such that $U\subseteq C$. We say $X$ is locally compact if this holds for all $x\in X$.

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Definition: Compactification

A compactification of a topological space $x$ is a pair $(X’,i)$ of a compact topological space $X’$ and an embedding $i:X\to X’$ such that $j(X)$ is a dense subset of $X’$, meaning every point of $X’$ is either in $X$ or a limit point of $X$.

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Theorem: One-point compactification

A topological space $X$ is locally compact and Hausdorff if and only if there exists a space $Y$ and an embedding $i:X\to Y$ such that:

• (i) $Y$ is compact and Hausdorff;
• (ii) $Y\backslash i(X)=\{p\}$, a single point;
• (iii) $(Y,i)$ is unique, in the sense that if any other $(Y’,i’)$ satisfies these properties, there exists a unique homeomorphism $f:Y\to Y’$ such that $f\circ i=i’$ (someday…commutative algebra diagram).

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Quotients §

Definition: Disjoint union

Given a collection of sets $\{X_{\alpha }{\}}_{\alpha \in J}$, their disjoint union, or coproduct, is

$$\coprod _{\alpha \in J}{X}_{\alpha }=\{(x,\alpha )|\alpha \in J,x\in X_{\alpha }\}.$$

For a single set $X$, while the usual union of $X$ with itself is simply the original set $X\cup X=X$, while $X\bigsqcup X$ involves two distinct copies of $X$.

Further, there exist natural injective inclusion maps $i_{\alpha }:X_{\alpha }\to {\coprod }_{\alpha }{X}_{\alpha }$ given by $x\mapsto (x,\alpha )$.

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Definition: Quotient map, relative quotient topology

A continuous function $p:X\to Y$ is a quotient map if $p$ is surjective and

$$U\subset Y\text{ open}⟺p^{-1}(U)\text{ open}.$$

The quotient topology on $Y$ (relative to $p$) is the unique topology such that $p$ is a quotient map.

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Lemma:

Let $p:X\to Y$ be continuous and surjective. Then $p$ is a quotient map if either of the following hold:

• (i) $p$ is an open or closed map (sends open sets to open sets, or closed sets to closed sets);
• (ii) $X$ is compact and $Y$ is Hausdorff (❓ compare to compactness & homeomorphisms?)

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Theorem (Munkres 22.2): Characterizing continuous functions out of quotient spaces

Let $p:X\to Y$ be a quotient map. Let $Z$ be a space and let $f:X\to Z$ be a function which is constant on $p^{-1}(z)$ for all $z\in Z$. Then there exists a unique $\overline{f}:Y\to Z$ such that $\overline{f}\circ p=f$.

Moreover:

• (i) The universal property of quotients. $\overline{f}$ is continuous if and only if $f$ is continuous;
• (ii) $\overline{f}$ is a quotient map if and only if $f$ is a quotient map.

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Theorem: The relationship between quotient maps and quotient spaces

Let $p:X\to Y$ be a quotient map. Define and equivalence relation $\sim$ on $X$ by setting

$$x_1\sim x_2⟺p(x_1)=p(x_2).$$

Then there exists a homeomorphism $\overline{p}:X/\sim \to Y$, and the diagram

commutes. That is, every quotient map is of the form $X\to X/\sim$.

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Countability and separation axioms §

Topological manifolds §

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• All manifolds are metrizable (embed in to R^N for some N)

Countability axioms §

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The fundamental group §

Homotopies and path concatenation §

Definition: Homotopic, homotopy

Let $I=[0,1]$, $X,Y$ be topological spaces, and $f,f’:X\to Y$ be continuous. We say $f$ is homotopic to $f’$ if there exists a continuous mapping $H:X\times I\to Y$ such that for all $x\in X$, $H(x,0)=f(x)$ and $H(x,1)=f’(x)$. We call $H$ a homotopy from $f$ to $f’$, and we write $f\simeq f’$.

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Definition: Path-homotopy

Given $I=[0,1$, two paths $f,f’:I\to X$, e.g., from $x_0\in X$ to $x_1\in X$, are path-homotopic, written $f{\simeq }_{p}f’$ if there exists a homotopy $H:I\times I\to X$ such that $H(0,t)=x_0$ and $H(1,t)=x_1$; that is, the endpoints are fixed for all $t\in I$.

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Scratchpad §

To do:

• Inclusion function
• Example objects used in topology, example topologies
  • ${\mathbb{R}}^{\omega }$
• Get examples of eveyrthing listed in lecture
• Neighborhoods?
• HW2 results: cts fns preserve limits
• HW3.1 the embeddings question
• EVT for Compactness
• Topologies on R – lower limit, finite complement

Additional notecards:

• To check topology is discrete, suffices to show every singleton is open (make its own note)

• What does an element of the product topology subbasis look like

  • What is the basis for the product/box topology

• The countably infinite product of the real line with itself

• Finish proof that (Theorem) The countable product of the real line is metrizable

• Account for the fact that homeomorphisms induce bijections between (path-)components

• Dense subsets and separatbility?