Moniker1998

There exist spaces which are Lindelof, pseudocompact and not compact, so we can't show that Lindelof implies realcompact without any assumptions.

Yeah, indiscrete topology on any set is an example here. Such space needs to be compact of course, a space is strongly connected and realcompact iff it's compact strongly connected...

A Tychonoff space X is easily seen to be strongly connected iff X has zero or one point. A strongly connected compact Hausdorff space needs to necessarily be Tychonoff, hence...

I'll try introducing more about realcompactness of spaces, including adding the Mysior plane, once the definition and basic theorems are here. For now it's postponed.

I'll add the property of realcompactness to all the spaces which don't have it this week. Of course I'll group them into different pull requests so it's easier to check....

Any ideas for how this space could be named on pi-base? It's related to Mrówka-Isbell spaces, such as https://topology.pi-base.org/spaces/S000057 Here's the construction. Fix a sequence of rationals converging to each...

Gillman and Jerison contribute this space to Katětov, so we could label it as Katětov's rational sequence topology space?

Some properties of $\Pi$: 1. Every subspace of $\Pi$ is extremally disconnected 2. Every subspace of $\Pi$ is $G_\delta$ 3. $\Pi$ is not first countable 4. $\Pi$ is not normal...

One thing to notice that $f:X\to 2^{2^\mathbb{R}}$ is a continuous bijection. However, I'm not sure if $2^{2^\mathbb{R}}$ is hereditarily realcompact.

My bad, pseudocompactness changes nothing in terms of realcompactness, so we still don't know.