Recently, a question was asked in MSE regarding homeomorphism in topology:

If two topological spaces have the same topological properties, are they homeomorphic?

I thought the question is not precise to me since OP does not what "topological properties" he is referring to. So I made a comment under the question:

What "topological properties" are you referring to?

Then I gave an answer:

A topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms.

If two topological spaces have the same topological properties, must they be homeomorphic?

One has to specify what set of topological properties he is referring to. For instance, consider the stand topology on $X=\mathbb{R}$ and subspace topology on $Y=[0,1]$. Note that "closed" "separability" is a topological property. Both $X$ and $Y$ are closed separable, but they are not homeomorphic.

The question in OP is vague and not precise. If you were saying "If two topological spaces are the same with respect to all the topological properties, must they be homeomorphic?" then the answer is YES, as it's been pointed out by @levap's answer.

I originally used "closedness" as a counterexample. Thanks to @Mariano Suárez-Álvarez's comment, I realized that "closedness" is not a property for topological spaces at all and I replaced it with "separability". However, the gentleman there insists that I am answering a different question from the one in OP and strongly opposes the way I understand the question in OP.

It is reasonable to say that I am not the only one who think the question in OP is ambiguous and not very clear, according to the votes of the first comment under OP. Also, in the very first sentence of @D_S's answer, it is pointed out that

For two arbitrary topological spaces $X$ and $Y$, there isn't a nice list of things you can check off on each of them to be able to say they are homeomorphic.

Note that OP did do some clarification in his/her editing. But as for the question in the title itself, I don't understand why my understanding is so "weird" that one would have such a strong objection.

What is wrong with my answer?

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    $\begingroup$ probably best not to answer questions you feel are poor, or poorly worded. There is not going to be any joy in it for you. $\endgroup$ – Will Jagy Oct 31 '16 at 3:45
  • $\begingroup$ @WillJagy: Thank you very much for your advice. Indeed, I just made a silly mistake. I will vote to close this thread and move on. $\endgroup$ – Jack Oct 31 '16 at 18:36

As a general rule, if you're not sure what the OP means in their question, or find something that is open for interpretation, it is always better to ask for clarification than to pose an answer that might not correspond with what the OP was asking. It is hard to produce good questions to poorly posed answers. In this case, you additionally posted an objectively incorrect answer first.

Note that you have stated that

One has to specify what set of topological properties he is referring to.

which assumed the OP had certain topological properties in mind. It is best if you comment, asking the OP for clarification on this.

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  • $\begingroup$ "It is hard to produce good questions to poorly posed answers." > "It is hard to produce good answers to poorly posed questions"? $\endgroup$ – Did Nov 2 '16 at 9:39
  • $\begingroup$ @Did Aha.${}{}{}$ $\endgroup$ – Pedro Tamaroff Nov 2 '16 at 13:29
  • $\begingroup$ Isn't this what you intended to say? $\endgroup$ – Did Nov 2 '16 at 13:40
  • $\begingroup$ @Did Y es. "Aha" means yes. $\endgroup$ – Pedro Tamaroff Nov 2 '16 at 13:41

I think there was/is an opportunity to produce a useful Answer surveying the non-trivial combinations of named topological properties that force (up to homeomorphism) a unique topological space.

A good bit of topology has this sort of motivation. For example, a topological continuum is a connected and compact Hausdorff space. What additional properties are needed to characterize the unit interval [0,1]?

The Answer you produced was substantially cogent, but retained a smattering of affect from your Comment that implied the Question had not been asked in a proper way. For future efforts I would suggest that it is better to pick an interpretation and run with it, while acknowledging that choosing poorly (or even wrongly) may evoke little positive response by the OP.

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