View source for Pointwise convergence, topology of

One of the topologies on the space $F(X,Y)$ of mappings from a set $X$ into a [[topological space]] $Y$. A [[generalized sequence]] $(f_\alpha)_{\alpha \in \mathfrak{A}}$ in $F(X,Y)$ converges pointwise to an $f \in F(X,Y)$ if $(f_\alpha(x))_{\alpha \in \mathfrak{A}}$ converges for any $x \in X$ to $x \in X$ in the topology of $Y$. The base of neighbourhoods of a point $f_0 \in F(X,Y)$ is formed by sets of the type $\{f : f(x_i) \in v_{f_0(x_i)} \,,\ i=1,\ldots n \}$, where $x_1,\ldots,x_n$  is a finite set of points in $X$ and $v_{f_0(x_i)} \in V_{f_0(x_i)}$ is a base of neighbourhoods at the point $f_0(x_i)$ in $Y$.

If $Y$ is a Hausdorff space, then $F(X,Y)$ is also Hausdorff and $A \subseteq F(X,Y)$ is compact if and only if it is closed and for every $x \in X$ the set $A_x = \{ f(x) : f \in A \}$ is compact.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Kelley,   "General topology" , Springer  (1975)</TD></TR>
</table>



====Comments====
There has been a lot of research into the interplay between the topological properties of [[Tikhonov space|Tikhonov]] (i.e., completely regular) spaces $Y$ and the topological (or linear topological) properties of $C_p(Y)$, where $C_p(Y)$ is the space of continuous real-valued functions on $Y$, endowed with the topology of pointwise convergence: there is the fundamental result of J. Nagata that two Tikhonov spaces $X,Y$ are homeomorphic if and only if the topological rings $C_p(X)$ and $C_p(Y)$ of continuous real-valued functions on $X$ and $Y$, respectively, with the topology of pointwise convergence are topologically isomorphic. See [[#References|[a1]]], [[#References|[b1]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Arkhangel'skii,   "A survey of $C_p$-theory"  ''Questions &amp; Answers in Gen. Topol.'' , '''5'''  (1987)  pp. 1–109</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Engelking,   "General topology" , Heldermann  (1989)</TD></TR>
</table>

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