Prove that $f$ is a homeomorphism

continuitygeneral-topologymetric-spacesreal-analysis

Let $f: (M,d) \rightarrow (N,\rho)$ be a one-one and onto mapping. Prove that the following are equivalent:

  1. $f$ is a homeomorphism.

  2. $g: N\rightarrow \mathbb{R}$ is contiuous if and only if $gof: M
    \rightarrow \mathbb{R}$     is continuous.

To show $1 \implies 2$ I have proceeded in the following manner.

$f$ is a homeomorphism $\implies$ $f$ is continuous from $M \rightarrow N$ $\implies$ For any sequence $(x_{n})$ in $M$ converging to $x$ we have $f(x_{n}) \rightarrow f(x)$. Let, $g$ is continuous from $N \rightarrow \mathbb{R}$ Therefore, $g(f(x_{n}) \rightarrow g(f(x))$. Hence, $gof(x)$ is continuous in $M$.

Since, the statement 2 had an iff next, I assumed that $f$ is a homemomrphism and $gof$ is continuous and proved that $g$ is continuous.

Now, I am not sure how I have to show $2 \implies 1$

Source:Real Analysis by N.L Carothers Page 72, Problem 54

Best Answer

A hint for $\ 2\implies 1\ $ is to take $\ g(y) = \rho\left(y,f(x_0)\right)\ $ for any fixed $\ x_0\in M\ $, and show that the continuity of $\ g\circ f\ $ implies the continuity of $\ f\ $ at $\ x_0\ $.

Edit: I originally missed the OP's statement that the proof of the continuity of $\ g\ $ following from $1$ and the continuity of $\ g\circ f\ $ had already been done. So I was incorrect in my earlier statement that this was still needed to complete the proof of $\ 1\implies 2\ $.

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