Talk:Relatively compact subspace

Almost periodic function

The last sentence

"The definition of almost periodic function F is at a conceptual level to do with the translates of F being a relatively compact set. This needs to be made precise in terms of the topology used, in a particular theory."

could be rewritten in much better English, and in particular, the "is at a conceptual level to do" should definitely be changed.

Also, in a metric space, an alternative definition of relative compactness is that every sequence has a cauchy subsequence, no? It might be nice to mention that on the main page (unless it's not an equivalent definition, but I think it is). Lavaka 22:37, 13 November 2006 (UTC)[reply]

The condition that every sequence have a Cauchy subsequence is equivalent only if the ambient space is complete; in general, that condition gives you total boundedness. —;Toby Bartels (talk) 16:51, 25 August 2008 (UTC)[reply]

Is this another name for precompact? And is it true that compact operators are those that map compact sets to relatively compact ones? — Preceding unsigned comment added by 74.129.169.64 (talk) 09:59, 10 January 2018 (UTC)[reply]

Definition

There is a serious difficulty with the definition as given ("relatively compact = has compact closure"); in a non-Hausdorff space, we can have compact sets which are not relatively compact. In my view, a superior definition would be: Y \subseteq X is relatively compact in X if it is included in a compact subset of X . I don't like to make the change without consulting the author; any comments? David Fremlin (fremdh@essex.ac.uk), 26 September 2009.

However, it's not Wikipedia's place to make up definitions, and the "has compact closure" definition seems to be almost universal (though sometimes it means "has compact Hausdorff closure"). Very occasionally, the author even points out that compact sets need not be relatively compact. --Zundark (talk) 13:17, 16 February 2011 (UTC)[reply]

David Fremlin's suggestion is in fact Bourbaki's definition (General Topology, Chapter 1-4): relative compact:=included in a compact subset. For a Hausdorff space, this is of course equivalent to 'has compact closure'. 145.97.197.215 (talk) 01:36, 4 February 2012 (UTC)[reply]

There is even a further weaker definition of relative compactness that is used e.g. by Archangelskii, "General Topology II: Compactness, Homologies of General Spaces". A subset Y \subseteq X of a topological space X is relatively compact in X if every cover of the ambient space X contains a finite subfamily that covers Y. Equivalently, Y is relatively compact in X if every net in Y has a cluster point (in X) or equivalently, every net has a convergent subnet (with limit in X). We have the following relations between the different notions of relative compactness: "Y has compact closure" => "Y is contained in a compact subspace" => "every net in Y has a cluster point in X". Conversely, if Y is Hausdorff (or even a KC-space, i.e. compact subsets are closed) then we have the converse "Y is contained in a compact subspace" => "Y has compact closure". Moreover, if Y is T_3 (regular Hausdorff) then we have the converse "every net in Y has a cluster point in X" => "Y has compact closure". — Preceding unsigned comment added by Yadaddy ag (talk • contribs) 14:59, 3 February 2016 (UTC)[reply]

This looks like a great addition to the main page! 82.67.197.191 (talk) 08:39, 11 May 2017 (UTC)[reply]