Is projection of locally connected compact subset locally connected? This question was originally at Math Stackexchange, but had no answers:
https://math.stackexchange.com/questions/4348707/is-projection-of-locally-connected-compact-subset-locally-connected
Problem
Let $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$ be topological spaces, $Z = X \times Y$, $\mathcal{T}_Z$ be the product topology on $Z$, $f : Z \to X$ be defined by $f(x, y) = x$, and $C \subset Z$ be compact and locally connected.
Is $f[C]$ locally connected?
Background
A space $(Z, \mathcal{T}_Z)$, where $\mathcal{T}_Z$ is a topology on $Z$, is locally connected, if for each $z \in U \in \mathcal{T}_Z$ there exists a connected $V \in \mathcal{T}_Z$ such that $z \in V \subseteq U$.
A space is compact if every open cover has a finite subcover.
Locally connected subset whose image is not locally connected
The following shows that some restrictions are necessary for the subset $C$. Let $X = Y = \mathbb{R}$, and $Z' = \{(0, 1)\} \cup \{(1/n, 0) : n \in \mathbb{N}^{> 0}\}$. Then $Z'$ is locally connected, but not compact, and $f[Z'] = \{0\} \cup \{1/n : n \in \mathbb{N}^{> 0}\}$ is not locally connected.
Holds when $f\restriction C$ is a quotient map
Suppose $f\restriction C$ is a quotient map. Quotient maps preserve local connectedness. Therefore $f[C]$ is locally connected.
This question provides conditions for $f\restriction C$ being a quotient map. However, as shown there, $f\restriction C$ is not always a quotient map.
Non-quotient strategy
There exist maps which are continuous, surjective, and preserve local connectedness, but are not quotient; in the linked example $X$ and $Y$ are both locally connected. If the claim does hold, then a general solution to this problem may need a stronger theorem for preservation of locally connectivity which includes these maps.
 A: A counterexample to this question can be constructed as follows.
Let $X=[0,1]$ be the closed interval with the standard Euclidean topology.
Let $Y=\omega$ and $\mathcal T_Y$ be the topology on $Y$ consisting of the sets $W\subseteq Y$ satisfying two conditions:
$\bullet$ if $0\in W$, then $W=\omega$;
$\bullet$ if $1\in W$, then $\omega\setminus W$ is finite.
The definition of the topology $\mathcal T_Y$ implies that $\omega\setminus\{0,1\}$ is an open discrete subspace of $(Y,\mathcal T_Y)$.
It is easy to see that the space $(Y,\mathcal T_Y)$ is not locally connected at $1$.
Choose any sequence $(U_n)_{n=1}^\infty$ of pairwise disjoint nonempty open sets in $X=[0,1]$ and let $g:X\to Y$ be the function defined by
$$g(x)=\begin{cases}n&\mbox{if $x\in U_n$ for some $n\ge 1$};\\
0&\mbox{otherwise}.
\end{cases}
$$
The definition of the topology $\mathcal T_Y$ ensures that the function $g:X\to Y$ is continuous.
Let $C=\{(x,g(x)):x\in X\}\subseteq X\times Y$ be the graph of the function $g$. It is clear that $C$ is homeomorphic to $[0,1]$ and hence is compact, connected and locally connected. But the projection of $C$ onto $Y$ is not locally connected.
