If a surjective continuous map is factored through the canonical projection of the equivalence relation induced by that map then the yielded map is a continuous bijection

From Maths
Revision as of 22:32, 9 October 2016 by Alec (Talk | contribs) (Created page with "{{Stub page|grade=A|msg=Flesh out and then demote grade}} __TOC__ : '''Note: ''' "''Factoring a continuous map through the projection of an equivalence relation induced by t...")

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
Stub grade: A
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Flesh out and then demote grade
Note: "Factoring a continuous map through the projection of an equivalence relation induced by that map yields an injective continuous map" is an important precursor theorem

Statement

Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces and let [ilmath]f:X\rightarrow Y[/ilmath] be a surjective continuous map. Then [ilmath]f[/ilmath] can be factored through the canonical projection of the equivalence relation induced by [ilmath]f[/ilmath] to yield a continuous bijection[Note 1], [ilmath]\bar{f}:\frac{X}{\sim}\rightarrow Y[/ilmath][1].

Proof

We know already (from: "Factoring a continuous map through the projection of an equivalence relation induced by that map yields an injective continuous map" that [ilmath]\bar{f}:\frac{X}{\sim}\rightarrow Y[/ilmath] is injective and continuous.

Recall from passing to the quotient that if [ilmath]f[/ilmath] is surjective then so is [ilmath]\bar{f} [/ilmath] - we apply that here (we know we can factor as factoring is how we got [ilmath]\bar{f} [/ilmath] in the first place), thus [ilmath]\bar{f} [/ilmath] is surjective!

A surjective injection is of course called a bijection.

Grade: C
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
This could be more formal


Notes

  1. This may not be a homeomorphism (a topological isomorphism however! That would require its inverse was also continuous

References

  1. File:MondTop2016ex1.pdf