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One of the topologies on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p0732401.png" /> of mappings from a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p0732402.png" /> into a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p0732403.png" />. A [[Generalized sequence|generalized sequence]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p0732404.png" /> converges pointwise to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p0732405.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p0732406.png" /> converges for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p0732407.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p0732408.png" /> in the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p0732409.png" />. The base of neighbourhoods of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324010.png" /> is formed by sets of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324012.png" /> is a finite set of points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324014.png" /> is a base of neighbourhoods at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324015.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324016.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324017.png" /> is a Hausdorff space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324018.png" /> is also Hausdorff and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324019.png" /> is compact if and only if it is closed and for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324020.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324021.png" /> is compact.
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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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
There has been a lot of research into the interplay between the topological properties of Tikhonov (i.e., completely regular) spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324022.png" /> and the topological (or linear topological) properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324025.png" /> is the space of continuous real-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324026.png" />, 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 (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]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Arkhangel'skii,  "A survey of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073240/p07324027.png" />-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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<table>
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<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>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR>
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</table>
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{{TEX|done}}

Revision as of 17:43, 28 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. See [a1].

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=40080
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article