Definition talk:Locally Connected Space

Was the original definition incorrect? --prime mover (talk) 07:09, 11 March 2017 (EST)

It was correct, as you can see at Equivalence of Definitions of Locally Connected Space. I did not change it. I just added two common definitions, whose equivalence is not obvious. --barto (talk) 07:22, 11 March 2017 (EST)

Note the analogy with the definitions of locally path-connectedness. --barto (talk) 07:25, 11 March 2017 (EST)

It seems like the one with neighborhood bases is also called weakly locally connected. While weakly local connectedness at a point does not imply local connectedness at that point, it is true that everywhere weakly locally connected is equivalent to everywhere locally connected (as proven in Equivalence of Definitions of Locally Connected Space). --barto (talk) 07:33, 11 March 2017 (EST)

The original was:

$T$ is locally connected if it has a local basis consisting entirely of connected sets.

It has been changed to:

$T$ is locally connected if it has a basis consisting of connected sets.

which is not the same thing.

Or in fact what you seem to have specifically chagned it to was this:

$T$ is locally connected if each point has a local basis consisting entirely of connected sets.

which is not what it says in the citations that are cited.

Either they are not wrong, in which case they stay as they are, or they are (in your opinion) wrong, in which case the correct version is to be backed up with a reference in cited literature, and an explanation posted up explaining where the discrepancy comes in, and whether it is merely a difference in terminology (in which case you need to be able to back up your change with a cogent argument) or whether it is a genuine error in the source works, which is to be backed up with solid evidence. As it is we have a number of pages referring to the original definition of "locally connected" which may no longer be valid. --prime mover (talk) 08:21, 11 March 2017 (EST)

Oh, I see. I forgot. I added "each point has a" because it makes no sense to talk about a local basis of a space, so I assumed those words were missing. (See Definition:Local Basis)

Either the original definition should have been it has a basis ... or each point has a local basis ....

Thinking about it, the first seems more likely; that's also how the same soure defined locally path-connected. --barto (talk) 08:54, 11 March 2017 (EST)

Let me review the original sources (I have both in my library) and see how they are worded. Please bear with me, this will not be immediate. Your analysis of their specific wordings will be welcomed -- we can then put what we think they "ought" to say, with an explanation as to why, and whether what they actually say is either wrong or over-simplified.

As I say, I will get to this later, I'm currently in the midst of working on some historical stuff in Number Theory which I don't want to tear myself away from immediately. --prime mover (talk) 09:15, 11 March 2017 (EST)

In Steen&Seebach, 2nd ed., locally connected is defined using (global) basis (page 30). Can't tell about the 1st ed. --barto (talk) 09:23, 11 March 2017 (EST)

Yes, I have now gone back to my copy. I likewise have 2ed. And I see that when I wrote this page originally, (see history, see first instance of page dated May 2011) I used "basis" not "local basis". I don't know why I would have changed it.

I also see that Sutherland (my first Topology text, the one which was the course book for my course in Topology some years ago) actually has this:

A space $T$ is locally connected if given any $x \in U \subseteq T$, where $U$ is open in $T$, there exists a connected open set $V$ such that $x \in V \subseteq U$.

I see where the confusion has come in, but I cannot reconcile as to why, or under whose influence I was when I made that change.

I am not immediately in the right headspace to even check whether Sutherland's definition is consistent with the S&S definition, but I suspect it is, as we have an equivalence page somewhere around here on the citation thread. Would you be able to reconcile this mess? This would be welcomed. --prime mover (talk) 09:39, 11 March 2017 (EST)

Yes, I'd love to. I'll replace those citations to the appropriate definitions. --barto (talk) 10:20, 11 March 2017 (EST)