Let X , Y and Z be topological spaces. Let f : X &#x2192;<!-- → -->

Blaine Stein

Blaine Stein

Answered question

2022-05-08

Let X, Y and Z be topological spaces. Let f : X Y and g : Y Z such that f is one-to-one. Suppose g f and f 1 are continuous. Can we conclude that g f ( X ) is continuous? If not under what conditions does it hold?

My attempt:
We know that g = ( g f ) f 1 and the domain of f 1 is f ( X ). Since g f and f 1 are continuous, then g is continuous on f ( X ).
I'm not sure if I missed some technicalities on the domains/codomains.

Answer & Explanation

Marco Meyer

Marco Meyer

Beginner2022-05-09Added 16 answers

Your reasoning is correct, although let me formalize it better. To be more precise, consider f : X f ( X ), f ( x ) = f ( x ), which is a bijection (since f is injective). What you say is that f 1 is continuous.
Next if g f is continuous then so is g | f ( X ) f because these are literally equal functions.
Finally g | f ( X ) = ( g | f ( X ) f ) f 1 , and thus it is continuous as well.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?