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Trying to review topology for a prelim, I'm starting to wonder exactly what topological properties do quotient maps, usually given as $p: X \rightarrow Y$, preserve? I believe quotient maps preserve compactness and path connectedness, but I'm not sure how it can be proved. Also, is it true that quotient maps do not preserve simple connectedness and discreteness? If that is so, what would be some good counterexamples? I would appreciate any helpful input on this, thanks.

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Compactness, path-connectedness, connectedness are all properties preserved by surjective continuous maps. However, discreteness is preserved by quotient maps and not surjective maps in general. –  Zhen Lin Jan 12 '13 at 17:32
    
@ZhenLin: How do you actually show that discreteness is preserved under quotient maps? –  Libertron Jan 12 '13 at 17:43
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The quotient topology on $Y$ is the finest topology making $p : X \to Y$ continuous; if $p$ is a quotient map then by definition $Y$ has the quotient topology, and obviously the discrete topology on $Y$ makes $p : X \to Y$ continuous if $X$ is also discrete. –  Zhen Lin Jan 12 '13 at 18:19

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up vote 3 down vote accepted

It's a well known property of compact spaces that their continuous images are compact. You will probably have been taught this at some point or can find it in any standard introduction to topology text book. Proving this to yourself would be a good exercise in using the definition of compactness (which a lot of people struggle with initially).

The preservation of path connectedness should be fairly clear. If two points are connected by a path $\gamma\colon I\rightarrow X$ in $X$, the image of $\gamma$ in $Y$ is still a path, and as every point in $Y$ has a preimage in $X$ this proposition falls out rather easily.

That quotient maps do not preserve simply connected spaces is a special case of the more general phenomenon that the fundamental group of a space isn't preserved under quotient maps. The most obvious example would be in identifying the two end points of the unit interval $[0,1]$ which gives a circle ($[0,1]/{\sim}\cong S^1$). The interval is simply connected but the circle has the free group on one generator $\mathbb{Z}$ as its fundamental group.

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By definition quotient maps are continuous, so they preserve any topological property that is preserved by continuous maps, including compactness and connectedness. They are rather badly behaved in most other ways, however. For example, consider the map

$$p:[0,2]\to\{0,1,2\}:x\mapsto\begin{cases} 0,&\text{if }x=0\\ 1,&\text{if }0<x<2\\ 2,&\text{if }x=2\;, \end{cases}$$

where $Y=\{0,1,2\}$ is given the quotient topology: $U\subseteq Y$ is open iff $p^{-1}[U]$ is open in $[0,2]$. Clearly the open sets in $Y$ are $\varnothing,\{0,1\},\{1\},\{1,2\}$, and $Y$, so $Y$ has the particular point topology with $1$ as the distinguished point. $[0,2]$ is a compact metric space, so it’s hereditarily normal. $Y$, on the other hand, is $T_0$ but has none of the higher separation properties.

In fact a quotient map preserves the $T_1$ property iff its fibres are closed. (The fibres of a map are the inverses of singletons.) You’ll find a little more information on preservation and non-preservation of topological properties here.

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Also connectedness, but not simply connectedness: think of the map $[0,1] \longrightarrow S^1$ which identifies $0$ and $1$.

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