Difference between revisions of "Quotient topology"

Revision as of 12:54, 7 April 2015 (view source) Alec (Talk | contribs) m ← Older edit		Revision as of 07:36, 8 April 2015 (view source) Alec (Talk | contribs) m Newer edit →
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	===Theorems===		===Theorems===
	{{Begin Theorem}}		{{Begin Theorem}}
−	Theorem: The quotient topology, {{M|\mathcal{Q} }} is the largest topology such that the quotient map, {{M|p}}, is continuous	+	Theorem: The quotient topology, {{M|\mathcal{Q} }} is the largest topology such that the quotient map, {{M|p}}, is continuous. That is to say any other topology such on {{M|Y}} such that {{M|p}} is continuous is contained in the quotient topology
	{{Begin Proof}}		{{Begin Proof}}
	For a map {{M|p:X\rightarrow Y}} where {{M|(X,\mathcal{J})}} is a [[Topological space]] we will show that the topology on {{M|Y}} given by:		For a map {{M|p:X\rightarrow Y}} where {{M|(X,\mathcal{J})}} is a [[Topological space]] we will show that the topology on {{M|Y}} given by:
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	Also known as:		Also known as:
	* Identification space		* Identification space
		+
		+	==See also==
		+	* [[Open and closed maps]]
	==References==		==References==

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Revision as of 07:36, 8 April 2015

Note: Motivation for quotient topology may be useful

Definition of Quotient topology

If

$$(X,\mathcal{J})$$ is a topological space, $$A$$ is a set, and $$p:(X,\mathcal{J})\rightarrow A$$ is a surjective map then there exists exactly one topology $$\mathcal{J}_Q$$ relative to which $$p$$ is a quotient map. This is the quotient topology induced by $$p$$

Quotient map

Let $(X,\mathcal{J})$ and $(Y,\mathcal{K})$ be topological spaces and let $p:X\rightarrow Y$ be a surjective map.

$p$ is a quotient map^[1] if we have

$$U\in\mathcal{K}\iff p^{-1}(U)\in\mathcal{J}$$

That is to say

$$\mathcal{K}=\{V\in\mathcal{P}(Y)|p^{-1}(V)\in\mathcal{J}\}$$

Also known as:

• Identification map

Stronger than continuity

If we had $\mathcal{K}=\{\emptyset,Y\}$ then $p$ is automatically continuous (as it is surjective), the point is that $\mathcal{K}$ is the largest topology we can define on $Y$ such that $p$ is continuous

Theorems

This theorem hints at the Characteristic property of the quotient topology

Quotient space

Given a Topological space $(X,\mathcal{J})$ and an Equivalence relation $\sim$, then the map:

$$q:(X,\mathcal{J})\rightarrow(\tfrac{X}{\sim},\mathcal{Q})$$ with $$q:p\mapsto[p]$$ (which is a quotient map) is continuous (as above)

The topological space $(\tfrac{X}{\sim},\mathcal{Q})$ is the quotient space^[2] where $\mathcal{Q}$ is the topology induced by the quotient

Also known as:

• Identification space

See also

• Open and closed maps

References

1. Jump up ↑ Topology - Second Edition - James R Munkres
2. Jump up ↑ Introduction to topological manifolds - John M Lee - Second edition