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Definition:   topological space $\ X\ $ is   r-basic $\ \Leftarrow:\Rightarrow\ $ the interiors of retracts of $\ X\ $ form a topological base of $\ X.$

Main question: Are r-basic spaces mentioned in any literature?

Questions:

  1. Does there exist a Hausdorff space that is not r-basic? [closed]
  2. Does there exist a a non-metric r-basic Hausdorff space? [closed]
  3. Do connectivity components inherit the r-basic property? In other words, is there an r-basic space that is not locally connected?
  4. If so, what is the largest natural class of spaces known to you that are r-basic? [now: topological manifolds]

Answers

  1. Yes

At first any r-basic Hausdorff space is regularly because in a Hudsorff space, a retract is closed.

Second, let S be a property of topological spaces inherited by quotient spaces. Then any space that is S but not locally S is not r-basic. In particular, S = "linear connectivity", S = "connectivity" provide a many examples of non-r-basic spaces (KP Hart's answer)

  1. All zero-dimensional spaces, totally ordered sets with the order topology, and arbitrary products of r-basic spaces are r-basic, but these spaces are usually non-metrizable (JosephVanName's comments)
  2. [open]
  3. Topological manifolds is r-basic (implying Hausdorff and second countability, of course) Proof. For a given point, take a neighborhood whose closure is a closed ball. A closed ball is an absolute retract, therefore we have built a retract whose interior contains a given point.
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  • $\begingroup$ A zero-dimensional space is a Hausdorff space with a basis of clopen sets. The class of all zero-dimensional spaces satisfies your property. $\endgroup$
    – Joseph Van Name
    Nov 13, 2021 at 23:50
  • $\begingroup$ Conversely, if $X$ is a topological space and $Y=(X\times[0,1])/(X\times\{1\})$ is a contractible space, so every retract of $Y$ must also be contractible. However, you can easily select $X$ so that the interiors of the contractible subsets of $Y$ do not form a basis for the topology. $\endgroup$
    – Joseph Van Name
    Nov 13, 2021 at 23:55
  • $\begingroup$ @JosephVanName The class of zero-dimensional spaces is extremely small; it does not even include smooth manifolds. I'm not sure if the question is reading correctly (because I don't know English very well yet). I mean, I want to find the most general natural class of spaces in which the interiors of the retracts form the base of the topology. It would be great if this is true for all Hausdorff spaces, but I don't believe it. Is the same thing written in my question? $\endgroup$
    – Arshak Aivazian
    Nov 14, 2021 at 0:08
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    $\begingroup$ The English in the question looks fine. By the way, the sentiment that the class of zero-dimensional spaces is ' extremely small' is a matter of perspective. In set theoretic topology or with regards to Stone duality, all spaces that one looks at are zero-dimensional. $\endgroup$
    – Joseph Van Name
    Nov 14, 2021 at 2:38
  • $\begingroup$ @JosephVanName Yes, of course, this is a matter of perspective. I forget to mention that my perspective is differential and algebraic topology due to the fact that I'm used to reading the relevant forum topics, where everyone assumes this perspective by default. Thanks for checking my English :) $\endgroup$
    – Arshak Aivazian
    Nov 14, 2021 at 2:50

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For a general source of counterexamples: look at connected but not locally connected spaces. The retracts are connected but the neighbourhoods of some points are not. The Topologist's sine curve, Knaster's Bucket Handle, and the Pseudoarc are well-known examples.

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  • $\begingroup$ Great! Thank you! Your example is generalized to "for any property of topological spaces S inherited by quotient spaces, if X is S and r-basic, then X is locally S". It is interesting that for "S = connectedness", "S = linear connectedness" the space splits into a union of subspaces with the property S. Is it true that the connected components inherit this abundance of retracts? If so, then r-basic implies local connectivity, local lin. connectivity, etc. $\endgroup$
    – Arshak Aivazian
    Nov 14, 2021 at 15:53
  • $\begingroup$ By the way, the r-basic Hausdorff space is regular. I wonder if there is a non-metric r-basic Hausdorff space? $\endgroup$
    – Arshak Aivazian
    Nov 14, 2021 at 15:56
  • $\begingroup$ All zero-dimensional spaces, totally ordered sets with the order topology, and arbitrary products of r-basic spaces are r-basic, but these spaces are usually non-metrizable. $\endgroup$
    – Joseph Van Name
    Nov 14, 2021 at 18:52

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