3.02 Quotient topology: continuous maps

Below the video you will find accompanying notes and some pre-class questions.

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Notes

Maps from a quotient space

(0.14) Suppose that

$X$ is a space,

$\sim$ is an equivalence relation on

$X$ , and

$Y$ is another space. Given a map

$F\colon X\to Y$ we say $F$ descends to the quotient if there exists a map

$\bar{F}\colon X/\sim\to Y$ such that

$F=\bar{F}\circ q$ , where

$q\colon X\to X/\sim$ is the quotient map.

(1.47)

$F$ descends to the quotient if and only if

$F(x)$ depends on

$x$ only through its equivalence class

$[x]$ , that is if and only if

$x_1\sim x_2\Rightarrow F(x_1)=F(x_2).$

(3.07) Conversely, given a map $G\colon X/\sim\to Y$ , we can precompose with $q$ to get a map $F:=G\circ q\colon X\to Y$ . In other words, $\bar{F}=G$ . This means that:

Functions on the quotient space

$X/\sim$ are in bijection with functions on

$X$ which descend to the quotient.

Continuity of maps from a quotient space

(4.30) Given a continuous map

$F\colon X\to Y$ which descends to the quotient, the corresponding map

$\bar{F}\colon X/\sim\to Y$ is continuous with respect to the quotient topology on

$X/\sim$ . Conversely, given a continuous map

$G\colon X/\sim\to Y$ , the composition

$F=G\circ q\colon X\to Y$ is continuous and descends to the map

$\bar{F}=G$ on the quotient.

(6.48) For the converse, if

$G$ is continuous then

$F=G\circ q$ is continuous because

$q$ is continuous and compositions of continuous maps are continuous.

(7.33) If $F$ is a continuous map which descends to the quotient then, given an open set $V\subset Y$ , the preimage $\bar{F}^{-1}(V)$ is open in the quotient topology on $X/\sim$ if and only if $q^{-1}(\bar{F}(V))$ is open in $X$ (by definition of the quotient topology). But

$q^{-1}(\bar{F}^{-1}(V))=(\bar{F}\circ q)^{-1}(V)=F^{-1}(V)$ since

$F=\bar{F}\circ q$ . But

$F^{-1}(V)$ is open because

$F$ is continuous.

This will be extremely useful in future: to specify a continuous map on a quotient space

$X/\sim$ , we just need to specify a continuous map on

$X$ and check it descends to the quotient.

Pre-class questions

Let X be the space in the figure below (thought of as sitting inside R3) and let A be the red subset. Which of the following functions X→R descends to the quotient X/A?
- the projection to the $z$ -axis,
- the projection to the $x$ -axis,
- the projection to the $y$ -axis?

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Next video: 3.03 Quotient topology: group actions.
Index of all lectures.