Path components of quotient space

Question

Let $X$ be a topological space and $A \subseteq X$ a subspace. Is it true that $X/A$ is path-connected if and only if $A$ meets every path component of $X$? Intuitively this seems reasonable but I'm having a hard time writing out the details. Thanks for any help.

Answer 1

No, it's possible for $X/A$ to be path-connected even if $A$ doesn't meet every path component of $X$. Let $X$ be the following topologist's sine curve: $$ X = \left\{\left(x,\sin \frac 1 x\right): 0<x\le 1\right\} \cup \big( \{0\}\times[-1,1]\big), $$ and let $A = \{0\}\times[-1,1]$. Then $X$ has two path components, one of which is $A$, but $X/A$ is homeomorphic to $[0,1]$, which is path-connected.

(To prove that $X/A\approx [0,1]$, consider the map $q\colon X\to [0,1]$ given by $q(x,y) = x$. Because $X$ is compact and $[0,1]$ is Hausdorff, $q$ is closed and therefore a quotient map. Because $q$ makes the same identifications as the quotient map $X\mapsto X/A$, it follows that $X/A\approx[0,1]$.)