Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How could I show that metrizability is a topological property?

Well, this means that if I have a set X that is metrizable and a homeomorphic function f from X to Y, then I need to show that Y is metrizabke, correct?

If I let d be a metric in X? How do I construct a metric in Y using the bijection f?

share|cite|improve this question
up vote 4 down vote accepted

Hint: homeomorphism transport the structure of $X$ to the one in $Y$. Try to use that fact to "pull" the metric $d$ to some metric $\tilde d$ on $Y$ such that $f$ is now an isometry.

share|cite|improve this answer

Just let $\rho(y_1,y_2)=d\big(f^{-1}(y_1),f^{-1}(y_2)\big)$ and check (1) that $\rho$ is a metric on $Y$, and (2) that it generates the right topology.

Note that this should be the obvious thing to try. The fact that $X$ and $Y$ are homeomorphic means that from a topological point of view $Y$ is just $X$ under a different name, and the homeomorphism $f$ is the ‘translator’ from $X$ to $Y$. Thus, $f(x)$ should behave in $Y$ exactly as $x$ does in $X$, and if we can fit a certain distance $d(x_1,x_2)$ to two points of $X$ in a way that fits the topology of $X$, we ought to be able to fit the same distance between the points $f(x_1)$ and $f(x_2)$ in a way that fits the topology of $Y$.

share|cite|improve this answer
So how do I show that it generates the right topology? – Akaichan Mar 5 '13 at 6:04
On the other hand, I think I got it, thank you very much for your time. – Akaichan Mar 5 '13 at 6:10
@IvordesGreenleaf: You’re welcome. If you still have questions after you’ve thought about it more, feel free to ask. – Brian M. Scott Mar 5 '13 at 6:28

Hint: Try letting $d_Y(a,b)=d(f^{-1}(a),f^{-1}(b))$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.