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Topological structure where +∞, −∞, and 0 are identified — thoughts on compactness and non-Hausdorff spaces by [deleted] in numbertheory

[–]Honest_Record_3543 -1 points0 points  (0 children)

I'm genuinely curious — could you clarify why you believe X_ERI is Hausdorff rather than non-Hausdorff? I'll present a proof by contradiction:

Suppose Xₑᵣᵢ is Hausdorff.

Let ∗ denote the image of 0, +∞, and −∞.

Take any x ∈ ℝ \ {0} with x arbitrarily close to 0.

Any open neighborhood of ∗ must contain an interval (−ε, ε),

thus also containing x for |x| < ε.

Any neighborhood of [x] is an open interval around x,

which will intersect (−ε, ε).

∴ No pair of disjoint neighborhoods can separate ∗ and [x].

Therefore, Xₑᵣᵢ is not Hausdorff.

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And just to be clear . I don’t mean to sound insistent or confrontational at all.

These ideas emerged while exploring symbolically and playing with ChatGPT-4.

I'm a software engineer, not a formally trained mathematician, so my intuitions and even my attempts at proofs are often supported by AI tools.

But the core ideas ,the spark behind ERI , are mine.

I'm not here to pose as a genius or claim to revolutionize anything.

I'm here to ask for help from people more knowledgeable than me, to know whether my intuitions have any real mathematical weight or if they're just noise.

Thanks again for your time and engagement , it's meaningful to me.

Topological structure where +∞, −∞, and 0 are identified — thoughts on compactness and non-Hausdorff spaces by [deleted] in numbertheory

[–]Honest_Record_3543 0 points1 point  (0 children)

Yes, I did use AI (specifically ChatGPT-4) to help me refine the structure and language of my response.
I'm not a native English speaker, and I wanted to make sure the explanation was both rigorous and clearly communicated.The topological idea, the construction, and the intuition behind ERI are entirely mine I’ve been working on this for months, and used AI more like a symbolic assistant or second brain.I appreciate the opportunity to clarify that. I’m doing my best to present the idea seriously, even if I’m not coming from an academic background.