Universal covering spaces, semi-locally simply connected, etc...

[GeoffreySangston]

(I apologize if this has already been discussed. I looked around and couldn't seem to see anything about it. Also, I've been using pi-base pretty consistently since college and think it's an incredibly helpful tool, so I just want to say thanks!)

I don't see the following properties. Is there a good reason for omitting these?

• Contractible
• Locally contractible (on the same page as contractible)
• Strongly locally contractible (on the same page as contractible)
• Simply connected
• Semi-locally simply connected
• Existence of a universal cover (Corollary 82.2 of Munkres says this is equivalent to path connected + locally path-connected + semi-locally simply connected)
• Locally simply connected
• Acyclic

(Edit: I think it's safe to say the standard definitions for the following depend on having a basepoint, since they depend on statements about the higher homotopy groups. So maybe that is a justification for not having them.)

• Weakly contractible
• Being an Eilenberg-MacLane space
• Aspherical

And one could have properties of the fundamental group, in analogy to simply connected. For example, Hatcher has the following on page 49, "It is a theorem of [Shelah 1988] that for a path-connected, locally path-connected compact metric space X , π1(X) is either finitely generated or uncountable."

If these were added, pi-base should also have the Hawaiian earring space. (There does appear to be a controversy around this name. Hatcher uses the descriptive name The Shrinking Wedge of Circles.) And perhaps there should be the rose with infinitely many petals, which comes up when discussing the previous space. And I assume there's a whole canon of counterexamples here. I'd like to see spaces like the pseudocircle, but perhaps that requires being able to make a more complex kind of query than pi-base desires to support (e.g., specifying the homotopy groups in some complex way).

[StevenClontz]

I'll reply in more detail later, but we recently added the earring space: https://topology.pi-base.org/spaces/S000201 (and had similar conversations about the name). And we have https://topology.pi-base.org/spaces/S139 as well for comparison.

[StevenClontz]

But in general, any property preserved by homeomorphism that's of interest to researchers in general or set-theoretic topology is a good fit here.

[GeoffreySangston]

Oh I see. I suppose I need to clear my cache because https://topology.pi-base.org/spaces/S000201 gives
404
Property not found S000201
on Chrome. I opened it in Safari and it does show up.

So now I'm not sure which of those properties are already supported. I'll double check later.

[StevenClontz]

Use sync or reset at https://topology.pi-base.org/dev to get updates (we need to make it more visible).

[prabau]

@StevenClontz Can this "Discussion" be moved to an "issue" maybe?

[StevenClontz]

Now an issue at #818