Talk:Second-countable space

message: I think it would be nice to add to the Introduction an example of a topological space that isn't second countable.
* Just take \mathbb{R}\, with the discrete topology. A less trivial example would be an uncountable disjoint union of any space with at least one non-trivial open set. Albmont (talk) 22:02, 9 April 2008 (UTC)
thread_title:  Counterexample 

message: Is is true that second-countability is a necessary and sufficient condition for a manifold to embbed in some finite dimentional Euclidean space? 131.111.28.33 (talk) 15:06, 3 November 2008 (UTC)
thread_title:  Necessary and sufficient condition for a manifold to embbed in some finite dimentional Euclidean space? 

message: "Specifically, every second-countable space is separable (has a countable dense subset) and Lindelöf (every open cover has a countable subcover)."
Every second-countable space being separable is equivalent to countable choice, but does ZF prove that every second-countable space is Lindelof?
JumpDiscont (talk) 02:25, 13 October 2009 (UTC)
replies: text: Could you please ask this question at Wikipedia:Reference desk/Mathematics? There, I am sure that quite a few people will answer your question in great detail. You may add information to the article if you feel that some ideas are not explained in depth. --PST 03:57, 13 October 2009 (UTC)
thread_title:  Choice 

message: "Quotients of second-countable spaces need not be second-countable; however, open quotients always are."
What is meant by open quotient? Does this mean the equivalence classes used to make the quotient are all open? That would do it. I don't think this terminology is standard.  2001:468:D01:60:C62C:3FF:FE22:BEDC (talk) 00:29, 24 September 2013 (UTC)
thread_title:  "Open quotient" 

message: Under Examples is the following:
"Define an equivalence relation and a quotient topology by identifying the left ends of the intervals - that is, identify 0 ~ 2 ~ 4 ~ … ~ 2k and so on. X is second countable, as a countable union of second countable spaces. However, X/~ is not first countable at the coset of the identified points and hence also not second countable."
I am barely able to understand part of this and unable to understand the rest.  I'm not a mathematician.  I have never seen the word "identify" used the way it is here, and think it needs clarification.  I am not sure - is 'identity' the exact same as 'equivalent'?
One change I think would help would be "Define an equivalence relation ( ~ ) and..."
What would be the effect if instead of "...by identifying the left ends..." it said "by defining the left end of each interval as equivalent - that is: 0 ~ 2 ~ 4 ~ ... ~ 2k. ???
It sure makes it clearer to me.  Is there any value in using the term "identify" or is it unnecessary (and uncommon?) jargon.  ----------- I also do not understand at all the notation X/~  the forward slash throws me  - does this have anything to do with it being a quotient topology (which I have zero understanding of, another issue here).  Does this mean the set X with the equivalence relation?  Or something else?  ( X\~ would be without the equivalence relation or the quotient of the equivalence relation (however absurd that sounds to me) ?)
I understand two sets X\Y but not X/Y nor X\<operator>, nor X/<operator>.  Anyone besides me think this needs clarification?Abitslow (talk) 05:14, 9 November 2013 (UTC)
thread_title:  Examples need clarification.