Topological Spaces

Section 1 Topological Spaces

Definition 1.1.

Let \(a,b\in\mb R\text{.}\) The Euclidean open interval from \(a\) to \(b\) is the set

\begin{equation*} (a,b)=\setBuilder{x\in\mb R}{a\lt x\lt b} \end{equation*}

Definition 1.2.

Let \(x\in\mb R\) and \(S\subseteq\mb R\text{.}\) The point \(x\) is a Euclidean limit point of the set \(S\) if and only if for every Euclidean open interval \((a,b)\) containing \(x\text{,}\) there is a point \(y\in S\) such that \(x\not=y\) and \(y\in(a,b)\text{.}\)

Checkpoint 1.3.

Determine if each set has the number \(0\) as a limit point.

(a)

\(\mb Z\)

(b)

\(\mb R\setminus\mb Z\)

(c)

\(\setBuilder{\frac{1}{n+1}}{n\in\omega}\)

(d)

\(\mb Q\)

(e)

Any finite set \(F\subseteq\mb R\)

Definition 1.4.

A subset \(U\subseteq\mb R\) is called Euclidean open if and only if for every point \(x\in U\text{,}\) there exists a Euclidean open interval \((a,b)\) such that \(x\in(a,b)\subseteq U\text{.}\)

Checkpoint 1.5.

Determine if each set is Euclidean open or not Euclidean open.

\(\displaystyle [\pi,42)\)
\(\displaystyle (-3,-1)\cup(4,5.5)\)
\(\displaystyle \setBuilder{x}{2x+1>5}\)
\(\displaystyle \mb Z\)
\(\displaystyle \mb R\setminus\mb Z\)
\(\displaystyle \mb Q\)
A finite set \(F\subseteq\mb R\)

Theorem 1.6.

A subset \(U\subseteq\mb R\) is Euclidean open if and only if there exists a collection of Euclidean open intervals \(\mc U\) such that \(U=\bigcup\mc U\text{.}\)

Proposition 1.7.

Let \(x\in\mb R\) and \(S\subseteq\mb R\text{.}\) The point \(x\) is a limit point of the set \(S\) if and only if for every Euclidean open set \(U\) containing \(x\text{,}\) there is a point \(y\in S\) such that \(x\not=y\) and \(y\in U\text{.}\)

Theorem 1.8.

The Euclidean open subsets of \(\mb R\) satisfy the following properties.

\(\emptyset\) and \(\mb R\) are Euclidean open sets.
If \(\mc U\) is a collection of Euclidean open sets, then \(\bigcup\mc U\) is also a Euclidean open set.
If \(U,V\) are Euclidean open sets, then \(U\cap V\) is also a Euclidean open set.

Definition 1.9.

Let \(X\) be a set, and let \(\mc T\subseteq \mc P(X)\) satisfy the following properties.

\(\emptyset,X\in\mc T\text{.}\)
If \(\mc U\subseteq\mc T\text{,}\) then \(\bigcup\mc U\in\mc T\text{.}\)
If \(U,V\in\mc T\text{,}\) then \(U\cap V\in\mc T\text{.}\)

Then \(\mc T\) is called a topology on \(X\text{,}\) the pair \(\tuple{X,\mc T}\) is called a topological space, and elements \(U\in\mc T\) are called open sets of the space. (Usually \(\tuple{X,\mc T}\) is abbreviated to just \(X\) when the topology is known from context.)

Definition 1.10.

Let \(\mc T\subseteq\mc P(\mb R)\) be the collection of Euclidean open subsets of \(\mb R\) defined by Definition 1.4. Then by Theorem 1.8, \(\mc T\) is a valid topology for \(\mb R\) called the Euclidean topology.

Theorem 1.11.

Let \(X\) be any set. Then the following sets are topologies on \(X\text{.}\)

\(\mc T=\mc P(X)\) is called the discrete topology.
\(\mc T=\{\emptyset,X\}\) is called the indiscrete topology.

Proposition 1.12.

Let \(\mc T\) be a topology, and let \(\mc U\subseteq\mc T\) be finite. Then \(\bigcap\mc U\in\mc T\text{.}\)

Proposition 1.13.

Let \(\mc T\) be the Euclidean topology. There exists a collection \(\mc U=\{U_n:n\in\omega\}\) such that \(\bigcap\mc U\not\in\mc T\text{.}\)

Definition 1.14.

Let \(a,b\in\mb R\cup\setList{-\infty,\infty}\text{.}\) The following are called intervals of real numbers.

\begin{equation*} (a,b)=\setBuilder{x\in\mb R}{a\lt x\lt b} \end{equation*}

\begin{equation*} [a,b)=\setBuilder{x\in\mb R}{a\leq x\lt b} \end{equation*}

\begin{equation*} (a,b]=\setBuilder{x\in\mb R}{a\lt x\leq b} \end{equation*}

\begin{equation*} [a,b]=\setBuilder{x\in\mb R}{a\leq x\leq b} \end{equation*}

Checkpoint 1.15.

Show that each of the following is an example of a topological space \(\tuple{X,\mc T}\text{.}\)

Let \(X=\mb R\) and \(\mc T=\setBuilder{(x,\infty)}{x\in\mb R} \cup\setBuilder{[x,\infty)}{x\in\mb R}\cup\setList{\emptyset,\mb R} \text{.}\)
Let \(X=\mb R\) and \(\mc T=\setBuilder{(x,y)}{ x,y\in\mb R\cup\setList{-\infty,\infty} \text{ and }x\lt 0\lt y }\cup\setList{\emptyset} \text{.}\)
Let \(X=\mb R\) and \(U\in\mc T\) if for each \(x\in U\text{,}\) there exists \(a,b\in\mb R\) such that \(x\in[a,b)\subseteq U\text{.}\)
Let \(X=\setList{0,1}\) and \(\mc T=\setList{\emptyset,\setList{0},X}\text{.}\)
Let \(X=\mb Z\text{,}\) \(E=\setBuilder{n\in\mb Z}{n\text{ is even}}\text{,}\) \(D=\setBuilder{n\in\mb Z}{n\text{ is odd}}\text{,}\) and \(\mc T=\setList{\emptyset,E,D,X}\text{.}\)

Definition 1.16.

Let \(\tuple{X,\mc T}\) be a topological space and let \(x\in X\text{.}\) The set \(N\subseteq X\) is called a neighborhood of \(x\) if and only if there exists an open set \(U\in\mc T\) such that \(x\in U\subseteq N\text{.}\)

Proposition 1.17.

A subset \(U\) of a topological space \(X\) is open if and only if \(U\) is a neighborhood of every point it contains.

Definition 1.18.

Let \(S\subseteq X\) be a subset of a topological space. The point \(x\) is a limit point of the set \(S\) if and only if for every neighborhood of \(U\) of \(x\text{,}\) there is a point \(y\in S\) such that \(x\not=y\) and \(y\in U\text{.}\)

Proposition 1.19.

The point \(x\in\mb R\) is a limit point of \(S\subseteq \mb R\) with the Euclidean topology according to Definition 1.2 if and only if it is a Euclidean limit point according to Definition 1.18.

Definition 1.20.

Let \(S\subseteq X\) be a subset of a topological space. Then \(S'\) is the set of all limit points of \(S\text{,}\) called the derived set of \(S\text{.}\)

Definition 1.21.

Let \(S\subseteq X\) be a subset of a topological space. Then \(\cl S=S\cup S'\) is called the closure of \(S\text{.}\)

Checkpoint 1.22.

Calculate \(\cl S\) for each of the following examples.

\(S=(-1,1)\subseteq\mb R\) where \(\mb R\) has the Euclidean topology.
\(S=(-1,1)\subseteq\mb R\) where \(\mb R\) has the discrete topology.
\(S=(-1,1)\subseteq\mb R\) where \(\mb R\) has the indiscrete topology.
\(S=\mb Z\subseteq\mb R\) where \(\mb R\) has the Euclidean topology.
\(S=\mb Q\subseteq\mb R\) where \(\mb R\) has the Euclidean topology.

Definition 1.23.

Let \(H\subseteq X\) be a subset of a topological space. Then \(H\) is called closed if and only if \(H=\cl H\text{.}\)

Theorem 1.24.

Let \(H\subseteq X\) be a subset of a topological space. Then \(H\) is closed if and only if the set \(X\setminus H\) is open.

Proposition 1.25.

The closed subsets of a topological space \(X\) satisfy the following properties.

\(\emptyset\) and \(X\) are closed sets.
If \(\mc H\) is a collection of closed sets, then \(\bigcap\mc H\) is also a closed set.
If \(H,L\) are closed sets, then \(H\cup L\) is a closed set.

Theorem 1.26.

(This space intentionally left blank to not mess up numbering. TODO: delete this after fall 2022.)

Definition 1.27.

Let \(S\subseteq X\) be a subset of a topological space. The point \(x\) is a boundary point of the set \(S\) if and only if for every neighborhood of \(U\) of \(x\text{,}\) both \(U\cap S\) and \(U\setminus S\) are non-empty.

Let \(\bd S\) collect all the boundary points of \(S\text{.}\)

Proposition 1.28.

Let \(a,b\in\mb R\text{.}\) Then \(\bd (a,b)=\bd (a,b]=\bd [a,b)=\bd [a,b]=\{a,b\}\) with respect to the Eudlidean topology.

Definition 1.29.

Let \(S\subseteq X\) be a subset of a topological space. The point \(x\) is a interior point of the set \(S\) if and only if there exists a neighborhood \(U\) of \(x\) such that \(x\in U\subseteq S\text{.}\)

Let \(\int S\) collect all the interior points of \(S\text{.}\)

Proposition 1.30.

Let \(U\subseteq X\) be a subset of a topological space. Then \(U\) is open if and only if \(U=\int U\text{.}\)

Definition 1.31.

Let \(S\subseteq X\) be a subset of a topological space. The point \(x\) is a exterior point of the set \(S\) if and only if there exists a neighborhood \(U\) of \(x\) such that \(x\in U\subseteq X\setminus S\text{.}\)

Let \(\ext S\) collect all the exterior points of \(S\text{.}\)

Definition 1.32.

A partition of a set \(X\) is a collection \(\mc P\) such that \(X=\bigcup\mc P\) and \(A\cap B=\emptyset\) for all \(A,B\in\mc P\) where \(A\not=B\text{.}\)

Proposition 1.33.

Let \(S\subseteq X\) be a subset of a topological space. Then \(\setList{\int S,\bd S,\ext S}\) is a partition of \(X\text{.}\)

Proposition 1.34.

Let \(S\subseteq X\) be a subset of a topological space. Then \(\cl S=\int S\cup\bd S=S\cup\bd S\text{.}\)

Checkpoint 1.35.

Let \(A\) be a subset of a topological space \(X\text{.}\) Prove or disprove the following.

\(\displaystyle \int\int A=\int A\)
\(\displaystyle \int\cl A=\int A\)
\(\displaystyle \bd\bd A=\bd A\)
\(\displaystyle \ext\ext A=\int A\)
\(\displaystyle \int\ext A=\ext A\)
\(\displaystyle \int\bd A=\emptyset\)
\(\displaystyle \cl\ext A=X\setminus\int A\)

Checkpoint 1.36.

Let \(A,B\) be subsets of a topological space \(X\text{.}\) Prove or disprove the following.

\(\displaystyle \int(A\cap B)=\int A\cap\int B\)
\(\displaystyle \int(A\cup B)=\int A\cup\int B\)
\(\displaystyle \bd(A\cap B)=\bd A\cap\bd B\)
\(\displaystyle \bd(A\cup B)=\bd A\cup\bd B\)
\(\displaystyle \cl(A\cap B)=\cl A\cap\cl B\)
\(\displaystyle \cl(A\cup B)=\cl A\cup\cl B\)

Definition 1.37.

Let \(Y\subseteq X\) for a topological space \(\tuple{X,\mc T}\text{.}\) Then the subspace topology for \(Y\) is given by \(\mc T_Y=\setBuilder{U\cap Y}{U\in\mc T}\text{.}\)

Proposition 1.38.

The subspace topology is a valid topology.

Definition 1.39.

The Cantor set is the subset \(C\subseteq\mb R\) defined by \(C=\bigcap_{n\in\omega} C_n\text{,}\) where \(C_0=[0,1]\) and

\begin{equation*} C_{n+1}=C_n\setminus\bigcup_{0\leq k\lt 3^n} \left(\frac{3k+1}{3^{n+1}},\frac{3k+2}{3^{n+1}}\right). \end{equation*}

This set is usually considered as a closed subset of the Euclidean line, or as a subspace of the Euclidean line.

Theorem 1.40.

The interior of the Cantor Set is empty.

Theorem 1.41.

The Cantor Set is uncountable.