Conversation started Oct 18, 2015 at 12:39.
Oct 18, 2015 12:39
in Mathematics, 1 hour ago, by Danu
Hi guys, does anyone have a canonical example of a Hausdorff (topological) space that is non-metrizable?
I think we have several threads about this on main: google.com/…
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Q: A compact Hausdorff space that is not metrizable

beatingdeadhorsesIs there an example of a compact Hausdorff space that is not metrizable? I was thinking maybe the space of continuous functions $f: X \rightarrow Y$ between topological spaces $X, Y$, might work, but I'm sure I'm missing some conditions.

And you can probably look at some weak or weak* topologies, if functional analysis is your thing: google.com/…
This link gives some reference.
And, of course, it is worth checking pi-base. Created by this guy.
in Mathematics, 2 mins ago, by Martin Sleziak
@Danu I would either go with some space derived from $\omega_1$ or with some weak or weak* topology. Whatever you feel more comfortable working with. You can find some posts on the main with some examples. I listed a few of them here.
I see that some examples were also linked here:
in Mathematics, 1 hour ago, by Alec Teal
http://chat.stackexchange.com/transcript/message/24782264#24782264 http://chat.stackexchange.com/transcript/message/24782262#24782262 FOR FUTURE REFERENCE
Oct 18, 2015 13:10
@MartinSleziak thanks, especially for the pi-base site. Never heard of it before, and it seems pretty awesome.
@Danu I have learned about the existence of that site from some answers here on MSE.
It lists the examples from this (famous) book:
Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) have defined a wide variety of topological properties. It is often useful in the study and understanding of abstracts such as topological spaces to determine that one property does not follow from another. One of the easiest ways of doing this is to find a counterexample which exhibits one property but not the other. In Counterexamples in Topology, Steen and Seebach...
But I guess some additional spaces were also added into the database.
Oct 18, 2015 14:07
@MartinSleziak This isn't quite accurate. Austin Mohr originally developed something he called "SpaceBook". About the same time James Dabbs had developed a different topological space tool (the original name of which I forget). That was off-line for quite a period of time during which it developed into π-Base. Austin Mohr seems to have abandoned his Spacebook in favor of π-Base.
 
1 hour later…
Oct 18, 2015 15:28
Thanks for the info, Arthur Fischer.
 
1 hour later…
Oct 18, 2015 16:46
I noticed that it was called spacebook and later pi-base. But I did not know the story behind it.
 
22 hours later…
Oct 19, 2015 14:59
@MartinSleziak It would probably have been better if I have linked to this:

Hausdorff non-metrizable spaces

Aug 4 '12 at 0:49, 52 minutes total – 73 messages, 4 users, 0 stars

Bookmarked 49 secs ago by Martin Sleziak

 
Conversation ended Oct 19, 2015 at 14:59.