Does homeomorphism preserve second countabity?

Question

This seems obviously true and i proved it, but i couldn't find this in googls, so i'm asking this to make sure.

Let $X,Y$ be topological spaces.

Let $H:X\rightarrow Y$ be a homeomorphism.

Is $Y$ second-countable if $X$ is second-countable?

Yes. And if you have proved it, you don't have to ask anymore. — Michael Greinecker, Jan 29 '14 at 16:04
@Michael Thank you. I asked this to make it sure 100%. Sorry if this interrupts you. — John. p, Jan 29 '14 at 16:05
@Martín-BlasPérezPinilla Unfortunately a topological property is defined to be a property of a space which is preserved under homeomorphism - you can't use that circular logic in a formal setting. — Dan Rust, Jan 29 '14 at 16:14
In my definition is a property dependent only of the topology (the family of open sets) of the space. — Martín-Blas Pérez Pinilla, Jan 29 '14 at 16:18
@Martín-BlasPérezPinilla That's a fairly meaningless definition. — Dan Rust, Jan 29 '14 at 16:26
You can define a property without explicit mention of open sets. — Martín-Blas Pérez Pinilla, Jan 29 '14 at 16:43

score 3 · Accepted Answer · answered Jan 29 '14 at 16:42

3

More generally, second- countability is preserved under open quotient maps. The same holds for first-countability.

answered Jan 29 '14 at 16:42

Stefan Hamcke

1 Answers1