Pointwise convergence

A type of convergence of sequences of functions (mappings). Let $f_n : X \rightarrow Y$, $n=1,2,\ldots$ where $X$ is some set and $Y$ is a topological space; then pointwise convergence means that for any element $x \in X$ the sequence of values $y_n = f_n(x)$, $n=1,2,\ldots$ converges in the space $Y$. The function $f : x \mapsto \lim_n y_n$ is then the pointwise limit of the sequence $(f_n)$. The definition extends to generalized sequences of functions and their values.

An important subclass of the pointwise-convergent sequences for the case of mappings between metric spaces (or, more generally, uniform spaces) is that of the uniformly-convergent sequences (cf. Uniform convergence).

Comments

A base for the topology of pointwise convergence on $C(X,Y)$, the space of continuous mappings from $X$ to $Y$, is obtained as follows. Take a finite set $K \subset X$ and for each $x \in K$ an open subset $V_x$ in $Y$ containing $f(x)$; for a given $f$ an open basis neighbourhood is: $\{ g \in C(X,Y) : g(x) \in V_x\ \text{for all}\ x \in K \}$. See also Pointwise convergence, topology of.

References

[a1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. 86 (Translated from Russian)