# Factoring a continuous map through the projection of an equivalence relation induced by that map yields an injective continuous map

From Maths

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## Contents

- This theorem is a minor extension of "
*factoring a function through the projection of an equivalence relation induced by that function yields an injection*" by simply considering continuity in addition.

## 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*continuous*map. Then factoring [ilmath]f[/ilmath] through the

*canonical projection*of the

*equivalence relation*induced by the mapping [ilmath]f[/ilmath] can not only be done, but in addition the map it yields, [ilmath]\bar{f}:\frac{X}{\sim}\rightarrow Y[/ilmath], is a continuous injection

^{[1]}.

Furthermore, if [ilmath]f:X\rightarrow Y[/ilmath] is surjective then so is [ilmath]\bar{f}:\frac{X}{\sim}\rightarrow Y[/ilmath] also, making [ilmath]\bar{f} [/ilmath] a bijection^{[Note 1]}

## Proof

**Overview: **

We know already (from *factoring a function through the projection of an equivalence relation induced by that function yields an injection*) that we can factor [ilmath]f[/ilmath] through [ilmath]\pi:X\rightarrow\frac{X}{\sim} [/ilmath]^{[Note 2]} to get a unique (as the canonical projection of the equivalence relation is surjective) map:

- [ilmath]\bar{f}:\frac{X}{\sim}\rightarrow Y[/ilmath]

Which is injective.

- We must show [ilmath]\bar{f} [/ilmath] is continuous

This is easy, simply:

Recall the characteristic property of the quotient topology:

topological spaces and let [ilmath]q:X\rightarrow Y[/ilmath] be a quotient map. Then

Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be ^{[2]}:- For any topological space, [ilmath](Z,\mathcal{ H })[/ilmath] a map, [ilmath]f:Y\rightarrow Z[/ilmath] is continuous
*if and only if*the composite map, [ilmath]f\circ q[/ilmath], is continuous

### Proof body

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Would be good to show, the key thing is:

- By the characteristic property of the quotient topology we see that:
- [ilmath]f:X\rightarrow Y[/ilmath] is continuous
*if and only if*[ilmath]\bar{f}:\frac{X}{\sim}\rightarrow Y[/ilmath] is continuous.

- [ilmath]f:X\rightarrow Y[/ilmath] is continuous

So automatically, [ilmath]\bar{f} [/ilmath] is continuous! It's that easy!

We know it's injective from the factoring part mentioned above.## Notes

- ↑ See:
*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* - ↑ Recall, for [ilmath]x,y\in X[/ilmath] we defined:
- [ilmath]x\sim y\iff f(x)=f(y)[/ilmath]

## References