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A topological space is locally connected if every point has a neighborhood basis of connected open subsets.

A property of topological spaces is termed hereditary, subspace-hereditary, if every subset of a topological space having the property also has the property (when the subset is endowed with the subspace topology).

  • The (Alexandroff) topological spaces in which any point has the least open neighborhood are hereditarily locally connected.

  • Another example of hereditarily locally connected topological space is $L=\mathbb N\cup\{\infty\}$ with opens $U\subseteq L$ upward closed subsets such that $U\neq\{\infty\}$. Note that this space is not an Alexandroff space, since $\infty$ does not have the least open neighborhood.

I wonder what classes of topological spaces have the hereditarily locally connectedness property. Is there any study available regarding hereditarily locally connectedness?

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    $\begingroup$ It looks like the answerer D.S. Lipham is using a different notion of "hereditary local connectedness" than we have in the question. Your notion requires all subspaces to be locally connected while the answerer simply needs for every connected subspace to be locally connected. $\endgroup$ May 10, 2023 at 20:39

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Let $X$ be a set, and let $\kappa$ be an infinite cardinal. Say sets in $X$ are closed if their cardinality is at most $\kappa$. (This class includes discrete spaces as you mentioned as well as spaces like the countable complement topology on the reals.)

Let $Y$ be a subspace. If $\kappa\geq|Y|$ then $Y$ is discrete, and thus locally connected.

If $\kappa\lt|Y|$, then the space is hyperconnected: all nonempty open sets intersect because if not, their complements would be two sets of size $\kappa<|Y|$ that cover $Y$. Furthermore, every open set is of size $|Y|$ (since if not, the open set and its complement would be two sets of size $<|Y|$ that cover $Y$) and is thus a copy of $Y$, showing local connectedness. Thus any space in this class is hereditarily locally connected.

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  • $\begingroup$ Thank you for the answer! Could you elaborate why do all nonempty open sets intersect in $Y$ and why all of them are of size $\kappa$? $\endgroup$ May 22, 2023 at 13:37
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    $\begingroup$ They're all of size $|Y|$, not $\kappa$. I've fixed that error and added the requested clarifications. $\endgroup$ May 22, 2023 at 22:06
  • $\begingroup$ Thank you a lot! Another quick follow up question. Do you see a counterexample of hyperconnected space which is not hereditarily locally connected space? $\endgroup$ May 24, 2023 at 7:37
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    $\begingroup$ Take the real line with an extra point. Nonempty sets are open if they are the union of an open subset of the reals with the singleton extra point. This space is hyperconnected but contains the reals as a subspace. $\endgroup$ May 25, 2023 at 12:33
  • $\begingroup$ Nice examples! It helped a lot to clean up the false beliefs also. $\endgroup$ May 26, 2023 at 17:44

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