# Ex: 1-7, relation between homeomorphism and continuity

Reference: Exercise: 1-7. Chapter 1. Topology by Hocking and Young.

Prove that a one-to-one transformation $f: S \rightarrow T$ of a space S onto a space T is a homeomorphism if and only if both f and $f^{-1}$ are continuous.

Proof:

Consider the following defintion:

Definition 1-1: A real point p is a limit point of a set X of real numbers if for any positive number $\epsilon$ there exists a point $x \in X$ such that $0 < |p-x| < \epsilon$.

Definition of homeomorphism: A homeomorphism of S onto T is a one-to-one transformation $f: S \rightarrow T$ which is onto and such that a point p is a limit point of a subset X of S if and only if $f(p)$ is a limit point of $f(X)$.

But, the above two definitions when combined mean the following: at least for the case of a Euclidean space: A real valued function $y=f(x)$ defined on an interval $[a,b]$ is continuous provided that if $a \leq x_{0} \leq b$ and $\epsilon >0$, then there is a number $\delta >0$ such that if $|x-x_{0}|<\delta$, x in $[a,b]$, then $|f(x) - f(x_{0})|<\epsilon$.

The above sub-case settles the proof for the Euclidean space $E^{1}$.

Now, for the more general transformation $f: S \rightarrow T$, consider definition 1-1 above and the following two theorems (both being equivalent definitions of continuous functions):

Theorem 1-6: Let $f: S \rightarrow T$ be a transformation of the space S into the space T. A necessary and sufficient condition that f be continuous is that if O is any open subset of T, then its inverse image $f^{-1}{O}$ is open in S.

Next, theorem 1-7: A necesssary and sufficient condition that the transformation $f: S \rightarrow T$ of the space S into the space T be continuous is that if x is a point of S, and V is an open subset of T containing $f(x)$, then there is an open set, U in S containing x such that $f(U)$ lies in V.

Clearly, the above settles the claim for a general transformation which is one-to-onto that it is a homeomorphism if and only if both f and $f^{-1}$ are continuous.

QED.