Section 3 Continuity & Homeomorphisms
Definition 3.1.
Let \(f:X\to Y\) be a function. For \(A\subseteq X\text{,}\) let \(f[A]=\setBuilder{f(x)}{x\in A}\text{.}\) For \(y\in Y\text{,}\) let \(f^\leftarrow(y)=\setBuilder{x\in X}{f(x)=y}\text{.}\) For \(B\subseteq Y\text{,}\) let \(f^\leftarrow[B]=\setBuilder{x\in X}{f(x)\in B}\text{.}\)
Definition 3.2.
Let \(X,Y\) be topological spaces with \(x\in X\text{,}\) and let \(f:X\to Y\) be a function such that for every neighborhood \(V\) of \(f(x)\text{,}\) there exists a neighborhood \(U\) of \(x\) such that \(f[U]\subseteq V\text{.}\) Then \(f\) is said to be continuous at the point \(x\text{.}\)
A function that is continuous at every point of its domain is called continuous.
Proposition 3.3.
A function \(f:X\to Y\) is continuous if and only if \(f^\leftarrow[V]\) is an open subset of \(X\) for every open \(V\subseteq Y\text{.}\)
Proposition 3.4.
Let \(X,Y\) be topological spaces.
The identity function \(\iota:X\to X\) defined by \(\iota(x)=x\) is continuous.
Let \(y\in Y\text{.}\) The constant function \(c_y:X\to Y\) defined by \(c_y(x)=y\) is continuous.
Every function whose domain is a discrete space is continuous.
Every function whose range is an indiscrete space is continuous.
Checkpoint 3.5.
Verify that each of the following functions are continuous with respect to the Euclidean topology at each real number where they are defined.
\(\displaystyle f(x)=|x|\)
\(\displaystyle f(x)=x^2\)
\(f(x)=g(x)+h(x)\) for \(g,h:\mb R\to\mb R\) continuous.
\(f(x)=g(x)h(x)\) for \(g,h:\mb R\to\mb R\) continuous.
\(\displaystyle f(x)=1/x\)
Theorem 3.6.
If \(f:X\to Y\) and \(g:Y\to Z\) are both continuous, then \(g\circ f:X\to Z\) is continuous.
Definition 3.7.
Let \(f:X\to Y\) be a bijection such that both \(f\) and its inverse \(f^{-1}\) are continuous. Then \(f\) is called a homeomorphism and \(X,Y\) are said to be homeomorphic.
Checkpoint 3.8. Properties preserved by continuous functions.
Determine if the following hold if \(f:X\to Y\) is a continous surjection. If not, determine if they hold if \(f\) is a continuous bijection. If not, show that they hold if \(f\) is a homeomorphism.
If \(U\subseteq X\) is open, then \(f[U]\subseteq Y\) is open.
If \(H\subseteq X\) is closed, then \(f[H]\subseteq Y\) is closed.
If \(x\) is a limit point of \(A\subseteq X\text{,}\) then \(f(x)\) is a limit point of \(f[A]\subseteq Y\text{.}\)
Proposition 3.9.
Every topological space is homeomorphic to itself.
Proposition 3.10.
If \(f:X\to Y\) and \(g:Y\to Z\) are both homeomorphisms, then \(g\circ f:X\to Z\) is a homeomorphism.
Theorem 3.11.
Let \(a\lt b\) and \(c\lt d\) be real numbers. Then \((a,b)\) and \((c,d)\) are homeomorphic subspaces of the Euclidean line.
Theorem 3.12.
Let \((-1,1)\) and \(\mathbb R\) have their Euclidean (subspace) topologies. Then the function \(f:(-1,1)\to\mb R\) defined by \(f(x)=\frac{2x}{1-x^2}\) is a homeomorphism.
Theorem 3.13.
Let \(\mc B\) be a basis for the Euclidean topology on \(\mb R\text{.}\) Give \(K=\mb R\cup\setList{-\infty,\infty}\) the topology generated by the basis \(\setBuilder{\{-\infty\}\cup(-\infty,x)}{x\in\mb R}\cup
\setBuilder{(x,\infty)\cup\{\infty\}}{x\in\mb R}\cup\mc B\text{.}\) Then \(K\) is homeomorphic to the subspace \([0,1]\) of the Euclidean line.
Proposition 3.14.
The real line with the Sorgenfrey topology generated by the basis \(\setBuilder{[a,b)}{a,b\in\mb R}\) is homeomorphic to the real line with the reverse Sorgenfrey topology generated by the basis \(\setBuilder{(a,b]}{a,b\in\mb R}\text{.}\)
Proposition 3.15.
A continuous bijection \(f:X\to Y\) is a homeomorphism if and only if for each \(U\subseteq X\) open, \(f[U]\subseteq Y\) is also open.
Example 3.16.
The topology on the real line generated by the basis \(\{[a,b]:a,b\in\mathbb R\}\) is equal (not just homeomorphic!) to another topology we've previously defined for \(\mathbb R\text{.}\)