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Intro to Topology

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.

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.
  1. \(\displaystyle f(x)=|x|\)
  2. \(\displaystyle f(x)=x^2\)
  3. \(f(x)=g(x)+h(x)\) for \(g,h:\mb R\to\mb R\) continuous.
  4. \(f(x)=g(x)h(x)\) for \(g,h:\mb R\to\mb R\) continuous.
  5. \(\displaystyle f(x)=1/x\)

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.
  1. If \(U\subseteq X\) is open, then \(f[U]\subseteq Y\) is open.
  2. If \(H\subseteq X\) is closed, then \(f[H]\subseteq Y\) is closed.
  3. If \(x\) is a limit point of \(A\subseteq X\text{,}\) then \(f(x)\) is a limit point of \(f[A]\subseteq Y\text{.}\)

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 \(\mathbb R\text{.}\)