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FYI, I'm asking about this, because in dynamical systems and chaos and fractals, etc. one has some pretty wacky, singular behavior, and I'm trying to figure out how some of the singular craziness there is can be understood with conventional topological axioms. The time evolution operator in dynamical
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I don't want to edit anything before the editor who added that paragraph is made aware of this. He unfortunately has the habit of quoting Narici at every opportunity, even when it's not warranted. It would be good if he realizes he is being excessive in this. After that, we can remove most of that
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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 " (See above.) Is there a reference for this? Perhaps this
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I see that you added a few examples to the
Examples section. But please do not introduce them if you are not sure. A few comments: (1) every topological group is completely regular, even the non-commutative groups. (2) Don't introduce a link to Tychonoff corkscrew, which does not even exist in
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The paragraph explaining the history of the concept in the lead is taken straight from Narici & Beckenstein. Now Narici & Beckenstein is a very fine book for functional analysis, but they are not historians. And in their historical commentaries they have the annoying habit of sometimes
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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) --
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The lede states that there are completely regular spaces that are not
Hausdorff; and the first talk topic above suggests that this is often the case for topological groups. Can explicit examples be given? For example, by naming some topological group that is not Hausdorff?
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Knowledge; and would need a reference... (3) There is no such thing as "the partition topology"; it should be "a partition topology". And the sentence as written is not even correct. I am tempted to revert all your changes, unless you want to discuss here first.
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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?
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The article is correct: every topological group is completely regular (with the
Knowledge conventions that neither "topological group" nor "completely regular" implies Hausdorff). This follows from the existence of a
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already mentions something about that in the section "Quotients and normal subgroups". (see condition for when a quotient group with quotient topology is
Hausdorff) (please don't repeat it in this article)
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For a trivial example, a space with the indiscrete topology is completely regular and non-Hausdorff. Any group becomes a topological group when given the indiscrete topology. That's completely regular and
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defined over a suitably defined domain. These are even examples of topological groups (actually topological vector spaces, defined by a seminorm), hence completely regular, but not
Hausdorff. The closure
600:(or the closure is taken) then the quotient space is Hausdorff, and "that's why we only study those." Of course, something similar would be true if we just forget about the group structure entirely. And
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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?
482:, 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?
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So I'm not seeing anything in particular that leads to something that would be completely regular, but isn't
Hausdorff. So, the most interesting counter-example so far is the
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introduced by
Tychonoff. In fact, if you look at the 1930 paper from Tychonoff, Tychonoff himself mentions in a footnote that the notion was introduced in 1925 by Urysohn.
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I didn't realize that N&B was not reliable for history. Now that I know, I'll stop using them for that. Thanks for informing me. Feel free to remove that paragraph.
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Also, there is absolutely no need to spell out all the various transliterations of the name
Tychonoff in this article. That belongs perfectly in the linked article
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Could you be so bold, as to edit the article, and provide the appropriate lede? Maybe add some short history section describing how things got to here?
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of the zero function consists of the measurable functions that are zero almost everywhere. So that could be the kind of
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to find a continuous real-valued function that separates two closed subsets. Apparently, this theorem won't work for
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on every topological group, because a topological space is uniformizable if and only if it is completely regular. --
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I'm pretty sure Császár uses this in his book General Topology, but I haven't got a copy to check at the moment. --
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which I guess is safe to add to this article. I was hoping for something more ...arcane, sophisticated. Oh well.
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twisting the truth to get a better story. In this particular case, the notion of commpletely regular space was
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but sadly, it does not state when a quotient space might be regular. Conversely, going back to groups, the
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is a uniform space. (...stuff about how to make G/H uniform...) The corresponding induced topology on
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Carefully with correcting from ambiguous to unambiguous - do your hit the proper case? Best regards
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is completely regular, but need not be Hausdorff in general. For example, see the function spaces
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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Please do not blindly reference N&B for historical stuff. They are not reliable for that.
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From the above, I conclude that the only way for G/H to not be Hausdorff is for it to not be
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https://topology.pi-base.org/spaces?q=completely%20regular%2B~t2%2BHas%20a%20group%20topology
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In other terms, this condition says that x and F can be separated by a continuous function.
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In other terms, this condition says that x and F can be separated by a continuous function.
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but is not completely regular would be nice to have. I'm not seeing one, just right now.
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never uses the word topology, except to say the strong operator topology is used.
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An example of a regular space that is not completely regular is the
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Examples of completely regular spaces that are not Hausdorff?
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As a uniform space, every commutative topological group is
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Examples of regular spaces that are not completely regular?
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Thanks for your understanding. I'll modify the paragraph.
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section rules out non-Hausdorff counterexamples quickly:
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with identity element 1, the following are equivalent:
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completely dashes our hopes. Near the bottom, it says
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Feel free to add an example to the Examples section.
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1324:Mid-priority mathematics articles
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115:Knowledge:WikiProject Mathematics
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263:there is a continuous function f
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338:23:13, 19 November 2023 (UTC)
311:Andrey Nikolayevich Tychonoff
289:21:10, 24 February 2012 (UTC)
109:and see a list of open tasks.
1319:C-Class mathematics articles
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831:{\displaystyle H\subseteq G}
792:For every topological group
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158:Examples and Counterexamples
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1174:{\displaystyle G/H}
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59:Mathematics
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712:Kolmogorov
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184:uniformity
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749:Tychonoff
726:Hausdorff
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615:, then
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139:on the
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770:{ 1 }
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1228:)
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