Let $X$ be a topological space. For $A \subseteq X$ we define the quotient space $X/A$ by the equivalence relation $$x \sim y \iff x = y \quad \text{or} \quad x,y \in A$$ and let $p : X \to X/A$ denote the associated quotient map.
Suppose $X$ is locally connected and compact topological space, show there exists a finite subset $A \subseteq X$ such that the quotient space $X/A$ is connected.
By local connectedness, each $x\in X$ has an open connected neighbourhoud $V_x$. Then, $\{V_x \mid x\}$ is an open cover of $X$ and by compactness it has an open subcover $V_{x_1}, \ldots, V_{x_n}$. If we define $A = \{x_1, \ldots, x_n \}$, the associated quotient space $X/A$ is equal to $$X/A = p(X) = p(V_{x_1}) \cup \ldots \cup p(V_{x_n}).$$ Since each $p(V_{x_j})$ is the continuous image of a connected set, they are connected in $X/A$. Hence, $X/A$ is the union of connected sets $p(V_{x_j})$. Since they all have a point in common, namely $A \in X/A$, the space is connected.