Isometries and Homeomorphisms

Isometry

Suppose $X$ and $Y$ are metric spaces. Suppose that $f:X\to Y$ is a bijection such that $d_Y(f(x),f(y))=d_X(x,y)\text{ for all }x,y\in X$Then $f$ is called an isometry between $X$ and $Y$. It preserves the distance between points. And we say that $X$ and $Y$ are isometric.

Homeomorphism

Suppose $X$ and $Y$ are topological spaces. If $f:X\to Y$ is a bijection and both $f$ and $f^{-1}$ are continuous we say that $f$ is a homeomorphism and that $X$ and $Y$ are homeomorphic.

e.g.

• $[0,1]$ and $(0,1)$ are not homeomorphic because $[0,1]$ is compact but $(0,1)$ is not.
• $\mathbb{R}$ and ${\mathbb{R}}^2$ are not homeomorphic. Otherwise, if $f:\mathbb{R}\to {\mathbb{R}}^2$ is a homeomorphism, then it also gives a homeomorphism from $R\setminus \{0\}\to {\mathbb{R}}^2\setminus \{f(0)\}$. This is impossible, because $\mathbb{R}\setminus \{0\}$ is not connected while ${\mathbb{R}}^2\setminus \{f(0)\}$ is connected.

Proposition

If $f$ is a homeomorphism then $U$ is open in $X$ if and only if $f(U)$ is open in $Y$.

Proof $U$ is open in $X$ implies that $f(U)$ is open in $Y$ as $f^{-1}$ is continuous; $V$ is open in $Y$ implies that $f^{-1}(V)$ is open in $X$ as $f$ is continuous.

Topological Property

If some property $P$ of a metric space is such that if $(X,d)$ has property $P$ then so does every metric space that is homeomorphic to $(X,d)$ we say that $P$ is a topological property. More colloquially, these are properties that are only concerned with set-theoretic notions and/or open sets, rather than distances. e.g. Topological properties:

• $X$ is open in $X$.
• $X$ is finite; countably infinite; or uncountable.
• $X$ has a point such that $\{x\}$ is open in $X$ (an ‘isolated point’)
• every continuous real-valued function on $X$ is bounded. Properties that are not topological:
• $X$ is bounded.