Homeomorphisms on Topological Spaces Review

# Homeomorphisms on Topological Spaces Review

We will now review some of the recent content regarding homeomorphisms on topological spaces.

Let $X$ and $Y$ be topological spaces and let $f : X \to Y$.

• On the Open and Closed Maps on Topological Spaces page we first defined the concept of open and closed maps from one topological space to the next. We said that $f$ is an Open Map if for every open set $U$ from the topological space $X$ we have that the image $f(U)$ is an open set in the topological space $Y$. Similarly, we said that $f$ is a Closed Map if for every closed set $C$ of $X$ we have that the image $f(C)$ is closed in $Y$.
• We looked at a rather important theorem regarding open and closed maps on topological spaces. We saw that if $f$ is and open or closed map then $f^{-1} : Y \to X$ is necessarily a continuous map from $Y$ to $X$. This is because if $f$ is an open map and we set $g = f^{-1}$ so that $g : Y \to X$ then for every open set $U$ in $X$ we have that $g^{-1} (U) = f(U)$ is open in $Y$ which we are given by the definition of an open map, and this implies the continuity of $g= f^{-1}$. Similarly if $f$ is a closed map, for every closed set $C$ in $X$ we have that $g^{-1} (C) = f(C)$ is closed in $Y$ by the definition of a closed map, and this also implies the continuity of $g = f^{-1}$.
• On the Homeomorphisms on Topological Spaces page we looked at a special type of open map. We said that a bijective map $f : X \to Y$ is said to be a Homeomorphism between the topological spaces $X$ and $Y$ if $f$ is both open and continuous, or similarly, $f$ and $f^{-1}$ are both continuous maps. We said that $X$ and $Y$ are Homeomorphic if there exists a homeomorphism between them.
• We then began to look at a lot of properties that are preserved with homeomorphisms.
(1)
\begin{align} \quad f(\mathrm{int} (A)) = \mathrm{int} (f(A)) \end{align}
(2)
\begin{align} \quad f(\bar{A}) = \overline{f(A)} \end{align}
(3)
\begin{align} \quad f(A') = (f(A))' \end{align}
(4)
\begin{align} \quad f(\partial A) = \partial (f(A)) \end{align}
• Clearly homeomorphisms preserve a lot of the structure of topological spaces. On the First Countability under Homeomorphisms on Topological Spaces page we saw that if $f : X \to Y$ is a homeomorphism and if $X$ is first countable then $Y$ is also first countable. Equivalently, if $X$ and $Y$ are not both first countable then there exists no homeomorphism $f$ between these topological spaces.
• Similarly, on the Second Countability under Homeomorphisms on Topological Spaces page we saw that if $f : X \to Y$ is a homeomorphism and if $X$ is second countable then $Y$ is also second countable. Equivalently, if $X$ and $Y$ are not both second countable then there exists no homeomorphism $f$ between these topological spaces.