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

[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 $Y$ 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 [a1], [b1].

References

[a1]	A.V. Arkhangel'skii, "A survey of $C_p$-theory" Questions & Answers in Gen. Topol. , 5 (1987) pp. 1–109
[a2]	R. Engelking, "General topology" , Heldermann (1989)