Pointwise convergence, topology of
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 ![]() |
[a2] | R. Engelking, "General topology" , Heldermann (1989) |
Pointwise convergence, topology of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pointwise_convergence,_topology_of&oldid=40080