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Presumably there will be a problem talking about all the transfinite cardinals, but assuming that this can be got around; is there an interesting topology on it - say such that the limit cardinals are actually limit points?

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You know en.wikipedia.org/wiki/Order_topology or are you looking for something else? –  Martin Brandenburg Mar 12 '13 at 1:11
    
It makes more sense to talk about transfinite ordinals than cardinals, though. I also feel that this question is somewhat overly broad. –  Asaf Karagila Mar 12 '13 at 1:17
    
@Brandenburg: Yes I do, but it's not in this question. I'm more curious as to what other natural topologies are available if GCH is either asserted or denied. –  Mozibur Ullah Mar 12 '13 at 1:30
    
@Karaglia - this was the reason for focusing on cardinals. But I'm not overly familiar with this area which explains both the caution and broadness. –  Mozibur Ullah Mar 12 '13 at 1:31
    
@Mozibur: Order-wise the classes of cardinals and ordinals are isomorphic, so it's just easier to talk about the ordinals, but it's really all the same. Note, also, that even in the ordinals the limit cardinals are limit of sequences made entirely of cardinals. –  Asaf Karagila Mar 12 '13 at 8:40

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The order topology is such topology. The ordinals, and in particular cardinals, are well-ordered sets. This means they carry the order topology, and limit ordinals (or cardinals if we restrict to those) are exactly those which are limits of sequences which are not eventually constant.

It should be noted that on successor ordinals this topology is always compact, and on ordinals whose cofinality is uncountable, sequentially compact.

If we topologize the entire class of ordinals, and we can give a parameterized description of open sets. This topology is compact (although not compact if we allow class-sized covers) and also sequentially compact for every length of sequences. The limit ordinals, are exactly the non-isolated points. We usually prefer to topologize a particular ordinal rather than all the class of ordinals.

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