Kinda related: Set of branch points isn't discrete, but branch points are isolated?
Consider a closed and continuous map $f: X \to Y$ of any topological spaces.
Question: What are some conditions on $f, X$ and $Y$ such that $f$ maps discrete to discrete?
Eg If $X$ and $Y$ are locally compact and hausdorff, if they are locally compact and T1, if they are 'relatively locally finite Hausdorff spaces', if they are 'topological surfaces', etc.
It says here ('p. 342', which is p. 3 of the file):
every image of a discrete space under a closed continuous map is discrete
and
every image of a discrete space under an open continuous map is also discrete
- and also here (p. 2, in the proof of Theorem 17.6):
As $f$ is closed the image of a discrete set of points is discrete.
- I guess neither answers the above question with 'any conditions'. I'm trying to understand which properties are of $f, X$ and $Y$ are relevant here.
Assume that $X$ is a discrete space.
If $f: X \to Y$ is open, then for every $y \in f[X]$ we write $y=f(x)$ and so $\{y\}= f[\{x\}]$ is open in $f[X]$ and so $f[X]$ is discrete. So for openness it's immediate.
If $f:X \to Y$ is closed and onto, then for $\{y\}$ in $Y$ we can say that
$$Y\setminus \{y\} = f[X\setminus f^{-1}[\{y\}]]$$ and hence $Y\setminus \{y\}$ is closed and $\{y\}$ is open in $Y$.
So we don't even need continuity.