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  -theory" Questions & Answers in Gen. Topol. , 5 (1987) pp. 1–109 |
| [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