Property Suggestion: Topologically rigid

[felixpernegger]

Property Suggestion

A space is said to be topologically rigid provided its only automorphism is the identity.

Rationale

This property appears in
https://www.sciencedirect.com/science/article/pii/S0166864121001966
and
https://arxiv.org/pdf/1912.01994

We cited those papers in the past for traits.

I dont belive it appears in other papers (though maybe with the name omitted)

Relationship to other properties

Trivial ones:
Has multiple points + Top rogid implies not homogenous
Top. rigid implies t0

I suspect there might be some regarding euclidean spaces and hereditarily connected and maybe some hard theorems also (didnt think about this)

I do not think this property should be added (soon at least) for multiple reasons. With this issue I just want to have it recorded somewhere for the future

[mathmaster13]

A manifold is never topologically rigid unless it is a single point or empty. I think when you remove the Hausdorff axiom there's a chance for bad things to happen. Removing second countability should be less of a problem.

The property is somehow hereditary, at least for nice spaces, but I can't easily come up with what sets it's hereditary with respect to.

Also, why not? (I have no opinions on whether it's added or not.) And why put the issue here for a property we ought not to add?

[felixpernegger]

The property is somehow hereditary, at least for nice spaces, but I can't easily come up with what sets it's hereditary with respect to.

Nice spaces dont really have this property

Also, why not? (I have no opinions on whether it's added or not.)

Becuase its not so common and it likely doesnt have too many implications

And why put the issue here for a property we ought not to add?

Because I didnt wanna forget about it, and maybe in the future there i a place for it (as I wrote in the issue...)

[prabau]

Actually, we have considered this property in the past (when we started discussing the Golomb/Kirch spaces years ago in pi-base). It is something that has some interest, that people could be wondering about for some spaces. The two results by T. Banakh and collaborators for the Golomb space and the Kirch space are famous. The thing is that they are not so easy to verify, demand a lot of effort if one wanted to check it in detail.
But I don't see a problem adding it at some point in the future (no urgency though).