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Pointwise convergence, topology of

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One of the topologies on the space of mappings from a set into a topological space . A generalized sequence converges pointwise to an if converges for any to in the topology of . The base of neighbourhoods of a point is formed by sets of the type , where is a finite set of points in and is a base of neighbourhoods at the point in .

If is a Hausdorff space, then is also Hausdorff and is compact if and only if it is closed and for every the set is compact.

References

[1] J.L. Kelley, "General topology" , Springer (1975)


Comments

There has been a lot of research into the interplay between the topological properties of Tikhonov (i.e., completely regular) spaces and the topological (or linear topological) properties of , where is the space of continuous real-valued functions on , endowed with the topology of pointwise convergence. See [a1].

References

[a1] A.V. Arkhangel'skii, "A survey of -theory" Questions & Answers in Gen. Topol. , 5 (1987) pp. 1–109
[a2] R. Engelking, "General topology" , Heldermann (1989)
How to Cite This Entry:
Pointwise convergence, topology of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pointwise_convergence,_topology_of&oldid=15233
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article