A surjective local homeomorphism need not be a covering map

Given: The objects, hypotheses, and choice principles stated above.

Every covering map is a surjective local homeomorphism, and each of its fibres is discrete in the subspace topology. (Covering maps are surjective local homeomorphisms with discrete fibres).

For a covering $p:E\to B$, the cardinality of $p^{-1}(b)$ is locally constant as a function of $b\in B$. If $B$ is connected, all fibres are equinumerous. (The cardinality of a covering fibre is locally constant and is constant on a connected base).

The underlying set. Let $I$ be a set and let $X_i$ be a set for each $i\in I$. The disjoint union is ${⨆}_{i\in I}{X}_i:={\bigcup }_{i\in I}(X_i\times \{i\}),$ whose elements are the pairs $(x,i)$ with $i\in I$ and $x\in X_i$. For $j\in I$ the $j$-th canonical injection is ${\kappa }_j:X_j\to {⨆}_{i\in I}{X}_i,{\kappa }_j(x):=(x,j)\mathrm{.}$ The construction is what makes the word "disjoint" honest. Each ${\kappa }_j$ is injective (def-injection-surjection-bijection), since $(x,j)=(x^{\mathrm{\prime }},j)$ forces $x=x^{\mathrm{\prime }}$; the images ${\kappa }_j[X_j]=X_j\times \{j\}$ are pairwise disjoint, since the second coordinate determines $j$; and their union is the whole set. So no assumption that the $X_i$ are disjoint as sets is needed, and none is made: the tag $i$ separates the copies even when $X_i=X_{i^{\mathrm{\prime }}}$ for $i\mathrm{\ne }{i}^{\mathrm{\prime }}$. (The disjoint union (coproduct) ${⨆}_i{X}_i$ with the final topology of the canonical injections: a set is open exactly when each of its traces is).

Throughout, $\mathbb{R}$ is the complete ordered field (def-complete-ordered-field, def-ordered-field) with its order (def-real-order). A subset $I\subseteq \mathbb{R}$ is order-convex when $x,y\in I$ and $x\le z\le y$ imply $z\in I$, and the intervals of $\mathbb{R}$ are the nine listed forms, among them $[a,b]=\{x:a\le x\le b\}$ and $(a,b)=\{x:a<x<b\}$. (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).