If $X$ is compact and connected will the quotient space $X/\sim$ have any special properties given $\sim$is an equivalence relation? Especially, can it be Hausdorff?
Asked
Active
Viewed 151 times
3 Answers
0
Of course $X/\sim$ is also compact and connected as a continuous image of $X$.
In this question and its answer and comments you'll find a characterisation when $Y$ will be Hausdorff when $X$ is.
Henno Brandsma
- 242,131
0
Since a quotient map is, by construction, continuous, the image of a compact, connected space will be compact, though not necessarily Hausdorff. An example of this last is the quotient $\mathbb R/ \mathbb Q $ , with $\mathbb Q$ as additive subgroup..
gary
- 4,027
0
Quotient (or even an image) of a connected space is connected.
Quotient (or even an image) of a compact space is compact. Here compactness means "every cover has a finite subcover". Some authors define compact space to be Hausdorff, we don't assume that here.
Quotient of a Hausdorff space need not be Hausdorff, even when the initial space is both compact and connected, e.g. $[0,1]/\sim$ where $x\sim y$ iff $x=y$ or $x,y\in(0,1)$. The quotient space is homeomorphic to $\{a,b,c\}$ with topology $\big\{\emptyset, \{b\}, \{a,b\}, \{b,c\}, \{a,b,c\}\big\}$ which is not Hausdorff.
Especially, can it be Hausdorff?
Yes, it can, see this: $X/\sim$ is Hausdorff if and only if $\sim$ is closed in $X \times X$
freakish
- 42,851