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closed interval with countably many origins #675

Open Moniker1998 opened 4 weeks ago

Moniker1998 commented 4 weeks ago

Hello, I've recently been browsing pi-base, and could not find a good example of a space that would be pseudocompact, not compact, $T_1$, and exhaustible by compact sets.

The space I am addressing in this issue is such example, see this post https://math.stackexchange.com/a/4935609/476484

prabau commented 4 weeks ago

If you want to write a pull request for it, that would be great.

Moniker1998 commented 4 weeks ago

I forgot how to do this, and additionally I'm going to be off for a week, but if no one is going to write a pull request then I will try to do it.

prabau commented 4 weeks ago

This is a variation on Line with countably many origins, modified so that the part away from the origins becomes compact, to get pseudocompactness. You could modify your example to be a circle with countably many origins, so that it also becomes Locally n-Euclidean (a non-Hausdorff one-dimensional manifold); this would allow to automatically deduce other properties, instead of having to deal with the endpoints of the interval $[-1,1]$.