View source for Exercises:Mond - Topology - 1/Question 8

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==Section B==
===Question 8===
</noinclude>Suppose that {{M|f:X\rightarrow Y}} is a [[continuous function]] that is also [[surjective]], and let {{M|\sim}} denote the ''[[equivalence relation]]'' [[equivalence relation induced by a map|induced by {{M|f}}]] on {{M|X}}.

Show that if {{M|X}} is [[compact]] and {{M|Y}} is [[Hausdorff]] then {{M|\frac{X}{\sim} }} is [[homeomorphic]] to {{M|Y}}.

Which theorem of [[Group Theory (subject)|group theory]] does this resemble?
====Proof====
Recall the result of question 5:
* "''[[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]]''"
Using the linked theorem we instantly obtain:
* {{M|\bar{f}:\frac{X}{\sim}\rightarrow Y}} that is a ''[[continuous]]'' [[bijection]]
** We must show this is a [[homeomorphism]].

Suppose {{M|X}} is [[compact]], then {{M|\frac{X}{\sim} }} is compact also as [[the image of a compact set is compact]] and {{M|\pi:X\rightarrow\frac{X}{\sim} }} is continuous (see [[quotient topology]] for more details)

We then apply the "''[[compact-to-Hausdorff theorem]]''" which shows us that {{M|\bar{f} }} is actually a [[homeomorphism]]

This is similar to the ''[[first group isomorphism theorem]]'' (in the case the map is surjective) certainly in a [[Category Theory (subject)|categorical]] sense. The group theorem "factors through the kernel of the morphism" where as this "factors through the equivalence relation induced by the morphism" and they both yield an isomorphism (a homeomorphism is the term for a topological [[isomorphism (category theory)|isomorphism]]. Here are diagrams:
* For any ([[surjective]]) [[group homomorphism]] {{M|\varphi:G\rightarrow H}} (for arbitrary groups) we get a [[group isomorphism]], {{M|\bar{\varphi} }}. {{M|1=\xymatrix{ G \ar[r]^\varphi \ar[d]_\pi & H \\ {G/\text{Ker}(\varphi)} \ar@{.>}[ur]_{\bar{\varphi} } } }}, and,
* For any ([[surjective]]) [[continuous map]], {{M|f:X\rightarrow Y}} (for an arbitrary compact space {{M|X}}, and arbitrary [[Hausdorff]] space {{M|Y}}) we get a homeomorphism/topological isomorphism {{M|\bar{f} }}. {{M|1=\xymatrix{ X \ar[d]_\pi \ar[r]^f & Y \\ \frac{X}{x_1\sim x_2\iff f(x_1)=f(x_2)} \ar@{.>}[ur]_(.6){\bar{f} } } }}
In both instances the "denominator" of the quotient depends on the map, factoring (as in [[passing to the quotient (function)]] is involved, rather than [[passing to the quotient (topology)]] and we get an isomorphism. 

I hope this is the similarity that is expected.
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==Notes==
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==References==
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