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At the moment I am learning about connected, path connected and locally connected topological spaces. I came across several (seemingly) different definitions for locally connected spaces and would very much appreciate If someone could help me make sense of them. I know that similar questions has been asked on the site (for instance here, here, and here) but they all seem to focused on the differences between the following definitions

  1. For every $x\in X$ and every neighbourhood $U \subseteq X$ of $x$ there exists $V \subseteq U$ that is open and connected such that $x\in V$.
  2. For every $x\in X$ and every open neighbourhood $U \subseteq X$ of $x$ there exists $V \subseteq U$ that is open and connected such that $x\in V$.

Both of which are different from the definition I am trying to make sense of. My conflict is between the Wikipedia definition which is

  1. A topological space is locally connected if every point admits a neighbourhood basis consisting of open connected sets.

To the definition given by Lee (Introduction to topological manifolds - page $92$) which sums up definitions $1$ and $2$ more "compactly" as follows;

  1. A topological space $X$ is locally connected If it admits a basis of connected open subsets.

I understand that the wikipedia definition implies definition $4$ (which is equivalent to $1$ and $2$) - Assume the wikipedia definition and let $x\in X$. By the assumption there exists a neighbourhood basis $\mathcal{B}$ of $x$ consisting of open connected sets. Let $U\subseteq X$ be neighbourhood of $x$ (regardless if $U$ is open or not, because if it is not open it contains some open neighbourhood $V$ of $x$) by the definition of a neighbourhood basis it follows that there exists a open and connected $B \in \mathcal{B}$ such that $B \subseteq U$ (or $B \subseteq V$ in the case $U$ is not open).

As I am not sure whether definitions $1,2$ or $4$ imply definition $3$, I would like to verify that. If they do not imply $3$ then the obvious question is, what should I consider as the definition of a locally connected space? is any of them preferred in some sense?

Thanks a lot!

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    $\begingroup$ So you see that 1, 2 and 4 equivalent? $\endgroup$ Commented 13 hours ago
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    $\begingroup$ If $\mathcal{B}$ is a basis consisting of connected open sets, and $x \in X$, what can you say about $\{ U \in \mathcal{B} : x \in U\}$? $\endgroup$ Commented 13 hours ago
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    $\begingroup$ You can say more. What would you want to be able to say, to see that def. 4 implies def. 3? $\endgroup$ Commented 12 hours ago
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    $\begingroup$ Closer to the wording of def. 3, you'd like to say that $\{ U \in \mathcal{B} : x \in U\}$ is a neighbourhood basis of $x$ [consisting of open connected sets]. Can you? $\endgroup$ Commented 12 hours ago
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    $\begingroup$ 1, 2 ,4 are equivalent. I wanted to know whether you have questions concerning this equivalence $\endgroup$ Commented 11 hours ago

1 Answer 1

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The implication $4. \implies 3.$ is a consequence of the following fact:

Let $(X, \tau)$ be a topological space, and $\mathcal{B}$ a basis of $\tau$. For $x \in X$, let $\mathcal{B}(x) = \{ U \in \mathcal{B} : x \in U\}$. Then $\mathcal{B}(x)$ is a neighbourhood basis of $x$ (consisting of open sets).

If, per definition 4., we choose $\mathcal{B}$ consisting of open connected sets, then the $\mathcal{B}(x)$ are neighbourhood bases consisting of open connected sets, as required in definition 3.

The implication $3. \implies 4$ follows from the converse of the above fact:

Let $(X,\tau)$ be a topological space, and for every $x \in X$, let $\mathcal{B}(x)$ be a neighbourhood basis of $x$ consisting of open sets. Then $\mathcal{B} = \bigcup_{x \in X} \mathcal{B}(x)$ is a basis of $\tau$.

If, per definition 3., we choose a neighbourhood basis $\mathcal{B}(x)$ consisting of open connected sets for every $x \in X$, the union of these families is a basis consisting of open connected sets as required in 4.

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