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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.
Clemens Bartholdy
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John Hughes
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    The Hausdorff property isn't transferred, is it? Let $X$ be any Hausdorff space with at least two points, and let $Y$ be the same set with the trivial (indiscrete) topology. The identity map $f : X \to Y$ is a continuous bijection. Or have I misunderstood a definition? – Nate Eldredge Aug 22 '23 at 14:05
  • Separability is transferred, first countability is not. – Nate Eldredge Aug 22 '23 at 14:11
  • So if P is to be a transferred property, then for any set $X$ of a given cardinality, one of the following must be true: (1) every topology on X has property P; (2) no topology on X has property P; (3) the trivial topology on X has property P and the discrete topology does not. – Nate Eldredge Aug 22 '23 at 14:13
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    Being "transferred by bijective continuous maps" is equivalent to being "transferred by coarser topologies" in the sense that if property $P$ holds for a topological space $(X,\tau)$, it also holds for $(X,\tau')$ for any coarser topology $\tau' \subseteq \tau$. – Geoffrey Trang Aug 22 '23 at 14:30
  • It was good of you to revive this. I'm not sure why Tian deleted it – FShrike Aug 22 '23 at 16:32
  • On reflection, we should note that Tian was incorrect in claiming that Hausdorffness is transferred (and therefore my comment about arc-connectivity might be false); take any non trivial Hausdorff space $(X;\mathcal{T})$ and $\mathrm{Id}:(X;\mathcal{T})\to(X;\text{indiscrete topology})$ which is a continuous bijection – FShrike Aug 22 '23 at 16:35
  • Thanks, folks, for the revived interest. I'm pleased that others found this dark corner of the subject a worthwhile place to shed a little light. – John Hughes Aug 22 '23 at 17:14
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    Arcwise connectedness (defined as "any two points are connected by an embedded copy of $[0,1]$") does not transfer, since the trivial topology is not arcwise connected (it doesn't contain any embedded copy of $[0,1]$ at all). See https://mathworld.wolfram.com/Arcwise-Connected.html – Nate Eldredge Aug 22 '23 at 17:42
  • https://math.stackexchange.com/questions/3364/topological-properties-preserved-by-continuous-maps – Clemens Bartholdy May 22 '24 at 04:11
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    Does this answer your question? Topological properties preserved by continuous maps – Clemens Bartholdy May 22 '24 at 04:11
  • I don't think the related post suggested by Babu answers this question, because it lacks the condition of bijectivity. Clearly one could look at each suggested item there and ask "would bijectivity alter this answer?", but that's hardly a "this has already been answered" situation. – John Hughes May 22 '24 at 13:42
  • But see my edits to my answer, which I've changed to Community Wiki and accepted. – John Hughes May 22 '24 at 13:44

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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.

John Hughes
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    The topologist's sine curve is connected. But a simpler counterexample for disconnectedness would be a two-point set with the discrete and the trivial topology. – Nate Eldredge Aug 22 '23 at 14:33
  • D'oh! Of course it is. It's the classic example of "connected but not path-connected." I guess this shows what being a non-mathematician for 25 years will do to your brain. Thanks for correcting this, and for the better example. – John Hughes Aug 22 '23 at 17:12
  • Thanks, FShrike, for editing to remove my error. – John Hughes Aug 22 '23 at 21:19