FoldUnfold Table of Contents The Hausdorff Property under Homeomorphisms on Topological Spaces The Hausdorff Property under Homeomorphisms on Topological Spaces Recall from the Hausdorff Topological Spaces page that a topological space $(X, \tau)$ is said to be Hausdorff if for every distinct pair of points $x, y \in X$ there exists open neighbourhoods $U$ of $x$ and $V$ … [Read more...]

## Second Countability under Homeomorphisms on Topological Spaces

FoldUnfold Table of Contents Second Countability under Homeomorphisms on Topological Spaces Second Countability under Homeomorphisms on Topological Spaces Recall from the Homeomorphisms on Topological Spaces page that if $X$ and $Y$ are topological spaces then a bijective map $f : X \to Y$ is said to be a homeomorphism if it is continuous and open. Furthermore, if such a … [Read more...]

## First Countability under Homeomorphisms on Topological Spaces

FoldUnfold Table of Contents First Countability under Homeomorphisms on Topological Spaces First Countability under Homeomorphisms on Topological Spaces Recall from the Homeomorphisms on Topological Spaces page that if $X$ and $Y$ are topological spaces then a bijective map $f : X \to Y$ is said to be a homeomorphism if it is continuous and open. Furthermore, if such a … [Read more...]

## The Boundary of a Set under Homeomorphisms on Topological Spaces

FoldUnfold Table of Contents The Boundary of a Set under Homeomorphisms on Topological Spaces The Boundary of a Set under Homeomorphisms on Topological Spaces Recall from the Homeomorphisms on Topological Spaces page that if $X$ and $Y$ are topological spaces then a bijective map $f : X \to Y$ is said to be a homeomorphism if it is continuous and open. Furthermore, if … [Read more...]

## The Set of Accumulation Points under Homeomorphisms on Topological Spaces

FoldUnfold Table of Contents The Set of Accumulation Points under Homeomorphisms on Topological Spaces The Set of Accumulation Points under Homeomorphisms on Topological Spaces Recall from the Homeomorphisms on Topological Spaces page that if $X$ and $Y$ are topological spaces then a bijective map $f : X \to Y$ is said to be a homeomorphism if it is continuous and … [Read more...]

## The Closure of a Set under Homeomorphisms on Topological Spaces

FoldUnfold Table of Contents The Closure of a Set under Homeomorphisms on Topological Spaces The Closure of a Set under Homeomorphisms on Topological Spaces Recall from the Homeomorphisms on Topological Spaces page that if $X$ and $Y$ are topological spaces then a bijective map $f : X \to Y$ is said to be a homeomorphism if it is continuous and open. Furthermore, if such … [Read more...]

## The Interior of a Set under Homeomorphisms on Topological Spaces

FoldUnfold Table of Contents The Interior of a Set under Homeomorphisms on Topological Spaces The Interior of a Set under Homeomorphisms on Topological Spaces Recall from the Homeomorphisms on Topological Spaces page that if $X$ and $Y$ are topological spaces then a bijective map $f : X \to Y$ is said to be a homeomorphism if it is continuous and open. Furthermore, if … [Read more...]

## Homeomorphisms on Topological Spaces Examples 2

FoldUnfold Table of Contents Homeomorphisms on Topological Spaces Examples 2 Example 1 Homeomorphisms on Topological Spaces Examples 2 Recall from the Homeomorphisms on Topological Spaces page that if $X$ and $Y$ are topological spaces then a bijective map $f : X \to Y$ is said to be a homeomorphism of these two spaces if $f$ is also open and continuous, or equivalently, … [Read more...]

## Homeomorphisms on Topological Spaces Examples 1

FoldUnfold Table of Contents Homeomorphisms on Topological Spaces Examples 1 Example 1 Example 2 Homeomorphisms on Topological Spaces Examples 1 Recall from the Homeomorphisms on Topological Spaces page that if $X$ and $Y$ are topological spaces then a bijective map $f : X \to Y$ is said to be a homeomorphism of these two spaces if $f$ is also open and continuous, or … [Read more...]

## Homeomorphisms on Topological Spaces

FoldUnfold Table of Contents Homeomorphisms on Topological Spaces Homeomorphisms on Topological Spaces Recall from the Open and Closed Maps on Topological Spaces page that we said that if $X$ and $Y$ are topological spaces then a map $f : X \to Y$ is said to be an open map (or simply open) if for every open set $U$ of $X$ we have that $f(U)$ is open in $Y$. Similarly, $f$ … [Read more...]