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PaperclipPerfector  5d ago (text post)   26556 thread views

Culture War Roundup for the week of February 17, 2025

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I'm going to pre-face this by saying I absolutely do not have the maths for this, or at least I've forgotten the explanation my cousin gave.

It's been quite a while since I first asked him, and I don't have him at hand. I don't recall if he pointed out if this holds true if and only if the initial Hausdorff space is also locally compact. I think he might have done so, but I can't be sure. In that case, the error would be mine, because I was only looking for a yes or no answer.

It's an interesting quandary I find myself in, given that I have authoritative sources on both sides as well as LLMs disagreeing with each other. I fed the initial question into the best LLMs I have access to, and if they said yes, then I submitted your rebuttal. They acknowledge that you are correct, though there is some quibbling in the reasoning traces I am not qualified to judge.

I have had mixed results asking reasoning models to answer. The ones I am most satisfied with is:

Final Answer: No. The one-point compactification of a Hausdorff space is Hausdorff if and only if the space is locally compact Hausdorff. Since there exist Hausdorff spaces that are not locally compact, the one-point compactification of such spaces are not Hausdorff. Therefore, the one-point compactification of a Hausdorff space is not necessarily Hausdorff.

Courtesy of Gemini 2.0 Flash Thinking.

ChatGPT with reasoning on:

Conclusion: The one-point compactification of a Hausdorff space is Hausdorff if and only if is locally compact. If is not locally compact, then there exists at least one point in that does not have a compact neighborhood, and consequently, the separation of that point from fails in .

Thus, the answer is: The one-point compactification of a Hausdorff space is Hausdorff if and only if the space is locally compact.

I can't really preserve the formatting with just markdown, so anything Latex would be missing. I'd be happy to share other responses in an offsite text repository, since I don't want to clutter up the thread.

My tentative conclusion is that both you and my cousin are correct, and even LLMs are (most of the time). My error was in expecting a yes or no answer due to my fallible memory.

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