Note The following question was asked by @Tian Vlašić, who then closed it after several interesting comments had been made. I was interested enough that I wanted to revive it.
Let us say that a property $P$ of topological spaces is transferred by bijective continuous maps if for every pair $X,Y$ of topological spaces, the statements $P(X)$ and there exists a bijective continuous map $f: X \rightarrow Y$ implies the statement $P(Y)$.
It is clear that every property of topological spaces that is transferred by bijective continuous maps is a topological property.
I have noticed that the following well-known topological properties are in fact transferred by bijective continuous maps:
-
the cardinality of the underlaying set of the topological space
-
the cardinality of the topology of the topological space
-
Hausdorffness of the topological space (Wrong! See comments)
I was quite surprised that Hausdorffness of the topological space is a property of topological spaces transferred by bijective continuous maps (note that this is a stronger statement than the statement that Hausdorffness of the topological space is a topological property).
My question is the following. What are some other well-known topological properties that are transferred by bijective continuous maps?
Note Some comments (from F. Shrike) on this question already noted the following:
- Cardinality is pretty trivially transferred
- Compactness and (path-)connectivity are transported (even if you remove injectivity as an assumption). A curious question is whether or not arc-connectivity is transported
- Compactness and connectivity are easy from surjectivity, they have extremely well known proofs (and duplicates on this site). As for arc-connectivity, I have no idea if this is true. It's true if the domain is Hausdorff by a difficult theorem I don't know the proof of + your claim that Hausdorffness is transported.
Best Answer
This is merely a partial answer:
While path-connectedness is preserved by such maps, dis-connectedness and path-disconnectedness are not. If $X$ is the topologist's sine curve $$ \{(0,0)\} \cup \{(x, \sin(1/x)) \mid 0 < x \le 1\} $$ then it is path-disconnected. But $f: X \to Y : (x, y) \mapsto x$ is a bijective continuous map, and its image is the unit interval, which is both connected and path connected. Of course, the inverse map is not continuous.
@Babu has suggested this related answer, which has a bunch of useful starting points, although bijectivity is not assumed, so turning them into answers to this question requires further effort.
Still, I feel as if enough has been done here to "accept" this Community-Wiki answer.