Equivalence relation on a topological space

Question

Define a relation on a topological space $X$ by $x \sim y$ iff there is a connected subspace $A \subseteq X$ that contains both $x$ and $y$. Prove this is an equivalence relation on $X$.

I figured out reflexivity and transitivity. The one I am trying to do is symmetry. How does this sound?

Let $x, y \in X$. We want to show that if $x \sim y$, then $y \sim x$. Since $x \sim y$, there is a connected subspace $A$ containing $x$ and $y$. Then their is a connected subspace $y \sim x$ in $A$ containing $y$ and $x$. Thus, $x \sim y$ and $y \sim x$.

That's perfect. One could even omit the second-to-last sentence :-) — Stefan Hamcke, Mar 21 '13 at 23:51
Reflexivity and symmetry are really trivial. Only for transitivity one has to do something. — Martin Brandenburg, Mar 21 '13 at 23:52

score 2 · Answer 1 · answered Mar 21 '13 at 23:51

2

Yes.

$x\in A$ and $y\in A$ if and only if $y\in A$ and $x\in A$.

answered Mar 21 '13 at 23:51

Asaf Karagila

393,674

Equivalence relation on a topological space

1 Answers1