Recollection of basic definitions: We recall the basic definitions that a continous map of topological spaces $f : X \to Y$ is open if $f(U)$ is an open subset of $Y$ whenever $U$ is an open subset of $X$. A continous map is locally injective if for every $x \in X$ there is an open subset of $x$ in $X$ such that the restriction to $U$ denoted $f|_U$ is injective.
My question: Suppose that $f: X \to Y$ is an open map of topological spaces. Then under which hypothesis (if any) on $X$ and $Y$ is $f$ locally injective?
In the simple case when $X$ and $Y$ are both $\mathbb{R}$, I don't even see how to produce an open map which is not a homeomorphism, which suggests that perhaps it is atleast true for say topological manifolds.
a) The maps $p:X\to Y$ both open and locally injective you are looking for are, practically tautologically, the local homeomorphisms.
An alternative terminology (especially for us ze French) is that $X$ is an étalé space over $Y$.
They are very important and correspond to the historical definition of sheaves, before the definition of sheaves as functors took over (but see here).
b) The most common way of obtaining a local homeomorphism is to take a smooth map $f:X\to Y$ between differential manifolds and to check that its differntial $T_xf:T_xX\to T_xY$ is a vector space isomorphism for all $x\in X$ : this guarantees that $f$ will even be a local diffeomorphism.
c) A purely topological source for étalé spaces consists of covering spaces.
Indeed, every covering space is an étalé space, but of course not conversely:
The inclusion $j:U\hookrightarrow X$ of any open subset $U$ of a topological space $X$ into that space is a local homeomorphism, but practically never a covering space.
d) Finally , every local homeomorphism $f: \mathbb R\to \mathbb R$ is indeed injective and thus gives rise to a homeomorphism $f: \mathbb R\to f(\mathbb R)$.
This is illustrated by $f(x)=e^x$ and is a rather elementary result in calculus: see here.