Quotient topology

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 $$p$$

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The quotient topology is actually a topology

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TODO: Easy enough

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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 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

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Theorem: The quotient topology, $\mathcal{Q}$ is the largest topology such that the quotient map, $p$, is continuous. That is to say any other topology such on $Y$ such that $p$ is continuous is contained in the quotient topology

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For a map $p:X\rightarrow Y$ where $(X,\mathcal{J})$ is a Topological space we will show that the topology on $Y$ given by:

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

is the largest topology on $Y$ we can have such that $p$ is continuous

Proof method: suppose there's a larger topology, reach a contradiction.

Suppose that $\mathcal{K}$ is any topology on $Y$ and that $p:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})$ is continuous.

Suppose that $\mathcal{K}\ne\mathcal{Q}$

Let $V\in\mathcal{K}$ such that $V\notin \mathcal{Q}$

By continuity of $p$, $p^{-1}(V)\in\mathcal{J}$

This contradicts that $V\notin\mathcal{Q}$ as $\mathcal{Q}$ contains all subsets of $Y$ whose inverse image (preimage) is open in $X$

Thus any topology on $Y$ where $p$ is continuous is contained in the quotient topology

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 where $\mathcal{Q}$ is the topology induced by the quotient

Also known as:

• Identification space

See also

• Open and closed maps