{{Short description|Topological space where every sequence has a convergent subsequence}} In mathematics, a topological space <math>X</math> is '''sequentially compact''' if every sequence of points in <math>X</math> has a convergent subsequence converging to a point in <math>X</math>. Every metric space is naturally a topological space, and for metric spaces, the notions of compactness and sequential compactness are equivalent (countable choice suffices; see below). However, there exist sequentially compact topological spaces that are not compact, and compact topological spaces that are not sequentially compact.
== Examples and properties ==
The space of all real numbers with the standard topology is not sequentially compact; the sequence <math>(s_n)</math> given by <math>s_n = n</math> for all natural numbers ''<math>n</math>'' is a sequence that has no convergent subsequence.
On a first countable space, a sequence <math>x_n</math> has a convergent subsequence if and only if :<math>\bigcap_n \overline{ \{ x_m \mid m \ge n \} }</math> is nomempty. Indeed, a limit of a convergent subsequence is necessarily in the above intersection (this direction holds for any topological space). Conversely, if <math>x</math> is in the above intersection, then let <math>x \in \cdots \subset U_2 \subset U_1</math> be a countable neighborhood base at <math>x</math>. Then, inductively, choose integers <math>n_i > 0</math> such that <math>n_i</math> is a least integer with the property (1) <math>n_i > n_{i-1}</math> and (2) <math>x_{n_i} \in U_i</math>, which is possible since <math>\mathbb N</math> is a well-ordered set. Then <math>x_{n_j} \to x</math>.
A point in the above intersection is called a cluster point. Thus, for first countable spaces, the definition of a sequentially compact space is the same as saying that each sequence in the space has a cluster point.
If a space is a metric space, then it is sequentially compact if and only if it is compact (cf. {{section link|Heine–Borel theorem|Generalization}}).<ref>Willard, 17G, p. 125.</ref> Here is how to see this, using only the countable Choice. We have to show "sequentially compact" implies "compact". First, we note <math>X</math> is totally bounded, meaning for each <math>\epsilon > 0</math>, there is a finite cover of <math>X</math> consisting of open balls of radius <math>\epsilon</math>. Indeed, if it fails for some <math>\epsilon</math>, by countable Choice, choose a sequence <math>x_n</math> such that :<math>x_n \not\in B(x_1, \epsilon) \cup \cdots \cup B(x_{n-1}, \epsilon).</math> This sequence <math>x_n</math> has no convergent subsequence, a contradiction. It follows that <math>X</math> has a countable base. Hence, it is enough to show <math>X</math> is countably compact; i.e., each descending sequence <math>E_1 \supset E_2 \supset \cdots</math> of nonempty closed subsets has nonempty intersection. But this is clear since :<math>\emptyset \ne \cap_n \overline{\{ x_m \mid m \ge n \}} \subset \cap_n E_n</math> for some sequence <math>x_n</math> with <math>x_n \in E_n</math>. <math>\square</math>
The first uncountable ordinal with the order topology is an example of a sequentially compact topological space that is not compact. The topological product of <math>2^{\aleph_0}=\mathfrak c</math> copies of the closed unit interval is an example of a compact space that is not sequentially compact.<ref>Steen and Seebach, Example '''105''', pp. 125—126.</ref>
== Related notions == A topological space ''<math>X</math>'' is said to be limit point compact if every infinite subset of ''<math>X</math>'' has a limit point in ''<math>X</math>'', and countably compact if every countable open cover has a finite subcover. In a metric space, the notions of sequential compactness, limit point compactness, countable compactness and compactness are all equivalent (if one assumes the axiom of choice).
In a sequential (Hausdorff) space sequential compactness is equivalent to countable compactness.<ref>Engelking, General Topology, Theorem 3.10.31<br> K.P. Hart, Jun-iti Nagata, J.E. Vaughan (editors), Encyclopedia of General Topology, Chapter d3 (by P. Simon) </ref>
There is also a notion of a one-point sequential compactification—the idea is that the non convergent sequences should all converge to the extra point.<ref>Brown, Ronald, "Sequentially proper maps and a sequential compactification", J. London Math Soc. (2) 7 (1973) 515-522. </ref>
== See also ==
* {{annotated link|Bolzano–Weierstrass theorem}} * {{annotated link|Fréchet–Urysohn space}} * {{annotated link|Sequence covering maps}} * {{annotated link|Sequential space}}
==Notes== {{Reflist}}
==References==
* {{cite book | author = Munkres, James | author-link = James Munkres | year = 1999 | title = Topology | edition = 2nd | publisher = Prentice Hall | isbn = 0-13-181629-2 }} * Steen, Lynn A. and Seebach, J. Arthur Jr.; ''Counterexamples in Topology'', Holt, Rinehart and Winston (1970). {{ISBN|0-03-079485-4}}. *{{cite book | author=Willard, Stephen | title=General Topology | publisher=Dover Publications | year=2004 | isbn=0-486-43479-6}}
Category:Compactness (mathematics) Category:Properties of topological spaces
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