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Difference between revisions of "Pointwise convergence, topology of"

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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. See [[#References|[a1]]].
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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]]].
  
 
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Revision as of 17:11, 29 December 2016

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)
How to Cite This Entry:
Pointwise convergence, topology of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pointwise_convergence,_topology_of&oldid=40095
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article