Constructing continuous functions

Theorem

1. Constructing continuous functions

   Let $X,Y,Z$ be topological spaces.

   • If $f\colon X\to Y$ is such that $f(X)=\{y_0\}$ for $y_{0}\in Y$, then $f$ is continuous.
   • If $A\subseteq X$ is a subspace, the inclusion $i_{A}\colon A\to X$ is continuous.
   • If $f\colon X\to Y$ and $g\colon Y\to Z$ are continuous, then the composition $g\circ f$ is continuous.

2. Continuation

   • If $f\colon X\to Y$ is continuous and $A\subseteq X$ is a subspace, the restriction $f\mid_{A}\colon A\to Y$ is continuous.
   • If $f\colon X\to Y$ is continuous and $Z\subseteq Y$ is a subspace such that $f(X)\subseteq Z$, then $f\mid \colon X\to Z$ is continuous. If $Z$ is a space containing $Y$ as a subspace, then $f\colon X\to Z$ is continuous.
   • If $X=\cup U_{\alpha}$, where each $U_{\alpha}$ is open and $f\mid_{U_{\alpha}}\colon U_{\alpha}\to Y$ is continuous for each $\alpha$, then $f$ is continuous.

Pasting lemma

1. The pasting lemma

   Let $X=A\cup B$ be a space, with $A,B$ closed in $X$. Let $f\colon A\to Y$ and $g\colon B\to Y$ be continuous. Suppose that $f(x)=g(x)$ for every $x\in A\cap B$, and define $h\colon X\to Y$ by:

   $$h(x)= \begin{cases} f(x) & \text{if \(x\in A\)},\\ g(x) & \text{if \(x\in B\)}. \end{cases}$$

   Then $h$ is continuous.

2. Proof

Maps into products

1. Maps into products

   Let $f\colon A\to X\times Y$ be given by:

   $$f(a)=(f_{1}(a),f_{2}(a)).$$

   Then $f$ is continuous if and only if the functions:

   $$f_1\colon A\to X,\quad f_2\colon A\to Y$$

   are continuous.

2. Proof

Homeomorphisms

1. Homeomorphism

   A bijective and continuous map $f\colon X\to Y$ is a homeomorphism if the inverse map $f^{-1}\colon Y\to X$ is also continuous

2. Equivalently, the continuous and bijective map $f\colon X\to Y$ is a homeomorphism if and only if $U\subseteq X$ open implies that $f(U)\subseteq Y$ is open.

3. Embedding

   If $f\colon X\to Y$ is injective and the bijection $f\colon X\to f(X)$ (where $f(X)\subseteq Y$ is a subspace) is a homeomorphism, we say that $f$ is an embedding.

Exercises

1. If $x_{0}\in X$, show that the map $f_{x_{0}}\colon Y\to X\times Y$ given by $f_{x_{0}}(y)=(x_{0},y)$ is an embedding.
2. Let $a,b\in \mathbb{R}$ with $a<b$. Show that $(a,b)$ is homeomorphic to $(0,1)$.
3. Let $Y$ be an ordered set with the order topology, and $f,g\colon X\to Y$ be continuous. Show that:
   • $\{x\in X\mid f(x)\leq g(x)\}$ is closed in $X$.
   • the map $h\colon X\to Y$ given by $h(x)=\min\{f(x),g(x)\}$ is continuous.
4. Let $f\colon A\to B$ and $g\colon C\to D$ be continuous functions between topological spaces. Show that the map $f\times g\colon A\times C\to B\times D$ given by $(a,c)\mapsto (f(a),f(c))$ is continuous.