Talk:Tychonoff space

message: -- "Every topological group is completely regular."
Teach me if I'm wrong, but I think this only holds for Hausdorff topological groups, maybe even only for locally compact groups. Since the wikipedia page for topological groups doesn't require hausdorff, this example should be removed (just in the case I'm right, for sure) -- Roman3 (talk) 09:47, 12 July 2010 (UTC)
replies: text: The article is correct: every topological group is completely regular (with the Wikipedia conventions that neither "topological group" nor "completely regular" implies Hausdorff). This follows from the existence of a uniformity on every topological group, because a topological space is uniformizable if and only if it is completely regular. --Zundark (talk) 10:43, 12 July 2010 (UTC)
thread_title:  Examples and Counterexamples 

message: I corrected from
:X is a completely regular space if 
:given any closed set F and 
:any point x that does not belong to F, 
:then 
:for every y in F there is a continuous function f from X to the real line R
:such that f(x) is 0 and f(y) is 1.
:In other terms, this condition says that x and F can be separated by a continuous function.
(This was introduced in http://en.wikipedia.org/w/index.php?title=Tychonoff_space&diff=next&oldid=463867677
with saying clearer if the quantifiers are in the beginning. Yes, clearer but wrong in this case.)
To
:X is a completely regular space if 
:given any closed set F and 
:any point x that does not belong to F, 
:then 
:there is a continuous function f from X to the real line R
:such that f(x) is 0 and, for every y in F, f(y) is 1.
:In other terms, this condition says that x and F can be separated by a continuous function.
Carefully with correcting from ambiguous to unambiguous - do your hit the proper case?
Best regards 90.180.192.165 (talk) 21:10, 24 February 2012 (UTC)
thread_title:  What a shame 

message: The Tietze extension theorem can be applied to normal spaces to find a continuous real-valued function that separates two closed subsets. Apparently, this theorem won't work for regular spaces, because, if it did, then every regular space would automatically be completely regular. So why does this theorem break down for regular spaces? What is the insight, intuition for this? 67.198.37.16 (talk) 23:24, 19 November 2023 (UTC)
thread_title:  Relationship to the [[Tietze extension theorem]]? 

message: It is mentioned that  is an alternative notation for Tychonoff spaces.  This notation does not appear in any of the standard references.  So at least it is not a notation in general use even if some author in the past may have used it once.  Does anyone have a reference for this? PatrickR2 (talk) 20:51, 3 December 2023 (UTC)
replies: text: I'm pretty sure Császár uses this in his book General Topology, but I haven't got a copy to check at the moment. --Zundark (talk) 14:08, 4 December 2023 (UTC)
thread_title:  {{em|T<sub>π</sub> space}} 

message: The article states "every topological group is completely regular". I had modified this to read "every commutative topological group is completely regular", but this was reverted, with a statement that "... even the non-commutative groups [are completely regular]" (See above.) Is there a reference for this? Perhaps this should be obvious? It's just not obvious to me. I don't have any particular intuition for this. Is there some way to think about this that would reveal the correctness of this statement?  67.198.37.16 (talk) 00:50, 1 February 2024 (UTC)
thread_title:  Every topological group is completely regular?