0

Let $(X,T)$ compact space with the equivalence relation such that:

$C=\{(x,y)\in X \times X: x \ \mbox{is related with}\ y\}$ is closed in $X\times X$. First I don't know if I understand this definition well, two elements are related iff $(x,y)$ is closed?.

Now the questions are:

Show that if $A$ is a closed subspace of $X$ then the set $ D=\{ x:\ \exists a \in A:\ (x,a)\in C\}$ is closed. I think that this set is closed because if x is related with $a$, then the set of points of the form $(x,a)$ is closed, then is this set is closed for properties of closure and A is closed then the set of points $x$ is a union of every $x$ has to be closed.

Second one: Show that the quotient space $(X/ \sim, T(\sim))$ is compact. I need a hint because I don't know how to start to proceed this is compact space.

Martin Sleziak
  • 53,687
energy
  • 147
  • It's not that x is related to y iff closed. They are not specifying the relation, just giving the hint that it is closed. – ziggurism Nov 27 '16 at 15:59
  • A relation on $X$ is a subset (call it $C$) of $X \times X$; a pair $(a,b) \in X\times X$ is "related", iff $(a,b) \in C$. They're not telling you what has to happen for $a$ and $b$ to be related; they're just telling you that the relation $C$, as a subspace of $X \times X$, is closed. – Jon Warneke Nov 27 '16 at 16:02
  • 2
    For the compactness, well continuous image of compact set is compact – ziggurism Nov 27 '16 at 16:11
  • Okey then for the first question would be: this set is the set of $x$ such that exists an a in A st $(x,a)$ is in R, so for being A closed x must be closed, and this set is the union of all x so its closed. And for the second question I use the projection $X$ to the quotient $X / \sym$ that is continuos so the image of X compact which is the quotient is compact. – energy Nov 27 '16 at 16:21
  • 1
    Hint: if $x\notin D$, then for all $a\in A, (x,a)\notin C$.i.e. $(x,a)\in X\times X\setminus C$, so there is a nbhd $U\times V\subseteq X\times X$ s.t. $x\in \pi_1 (U\times V)=U$. For the second part, find an appropriate map. – Matematleta Nov 27 '16 at 16:37

1 Answers1

2

For the first part, consider the set $(X\times A)\cap C=\{(x,a)\in X\times A:(x,a)\in C\}$ which is closed in $X\times X$ being the intersection of two closed sets.

Now, $D$ is clearly the image of this set under the projection map $\pi_1:X\times X\to X$.

By this result, since $X$ is compact, the projection map is closed. Thus $D$ being the image of a closed set is closed.

For the second part, consider the map $X\to X/\sim$ given by sending any $x\in X$ to its corresponding equivalence class $[x]\in X/\sim$. This map is clearly surjective and is continuous as $X/\sim$ has the quotient topology. Since the image of a compact set under a continuous map is compact we have that $X/\sim$ is compact.

Community
  • 1
R_D
  • 7,312