By spaces I mean locally compact, $\sigma$-compact, connected, locally connected, Hausdorff topological spaces.
I need a reference (not a proof, I already have one -- or at least I think so; counterexamples are welcome) for the following fact. I believe it should be standard.
If $X$ is a non-compact space, then the Alexandroff one-point compactification of $X$ is homeomorphic to the quotient space $\mathfrak{F}X\big/\mathcal{E}(X)$, where $\mathfrak{F}X$ is the Freudenthal compactification of $X$ and $\mathcal{E}(X)$ is the set of ends of $X$.
Definitions of both $\mathfrak{F}X$ and $\mathcal{E}(X)$ are recalled below.
Recall that an end of a topological space $X$ is a function $$\varepsilon: \{K\subseteq X: K \text{ is compact }\} \to 2^X\smallsetminus\{\emptyset\}$$ such that $\varepsilon(K)$ is an unbounded (= of non-compact closure) connected component of $X\smallsetminus K$ and $\varepsilon(L)\subseteq \varepsilon(K)$ for all compact sets $K\subseteq L \subseteq X$. The set of ends of $X$ is denoted by $\mathcal{E}(X)$.
For $\varepsilon \in \mathcal{E}(X)$ and a compact set $K\subseteq X$ define $$N(\varepsilon,K)=(X\smallsetminus K) \cup \bigl\{\delta\in\mathcal{E}(X):\delta(K)=\varepsilon(K)\bigr\}.$$
Let $\mathfrak{F}X=X\cup \mathcal{E}(X)$ with topology given by the following bases of open neighbourhoods: for a basis of neighbourhoods of $x\in X$ we take a basis of neighbourhoods of $x$ in $X$; for $\varepsilon\in\mathcal{E}(X)$ the basis of neighbourhoods is $$\{N(\varepsilon,K):K\subseteq X \text{ is compact }\}.$$
