# 3.02 Quotient topology: continuous maps

## Video

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

- Previous video:
**3.01 Quotient topology**. - Next video:
**3.03 Quotient topology: group actions**. - Index of all lectures.

## 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 \(\mathbf{R}^3\)) and let \(A\) be the red subset. Which of the following functions \(X\to\mathbf{R}\) descends to the quotient \(X/A\)?

- the projection to the \(z\)-axis,
- the projection to the \(x\)-axis,
- the projection to the \(y\)-axis?

## Navigation

- Previous video:
**3.01 Quotient topology**. - Next video:
**3.03 Quotient topology: group actions**. - Index of all lectures.