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| − | A [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084620/s0846201.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084620/s0846202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084620/s0846203.png" /> (that is, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084620/s0846204.png" /> is not closed), then there is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084620/s0846205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084620/s0846206.png" /> of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084620/s0846207.png" /> that converges to a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084620/s0846208.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084620/s0846209.png" /> always implies that there is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084620/s08462010.png" /> of points from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084620/s08462011.png" /> that converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084620/s08462012.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084620/s08462013.png" /> is called a Fréchet–Urysohn space. | + | {{TEX|done}} |
| | + | A [[Topological space|topological space]] $X$ such that if $A\subset X$ and $A\neq[A]$ (that is, the set $A$ is not closed), then there is a sequence $x_k$, $k=1,2,\dots,$ of points of $A$ that converges to a point of $[A]\setminus A$. If $x\in[A]\subset X$ always implies that there is a sequence $x_k$ of points from $A$ that converges to $x$, then $X$ is called a Fréchet–Urysohn space. |
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Revision as of 07:45, 23 August 2014
A topological space $X$ such that if $A\subset X$ and $A\neq[A]$ (that is, the set $A$ is not closed), then there is a sequence $x_k$, $k=1,2,\dots,$ of points of $A$ that converges to a point of $[A]\setminus A$. If $x\in[A]\subset X$ always implies that there is a sequence $x_k$ of points from $A$ that converges to $x$, then $X$ is called a Fréchet–Urysohn space.
Sequential spaces form a coreflective subcategory (see Reflective subcategory) of the category of all topological spaces; the coreflection is obtained by retopologizing an arbitrary space with the topology in which a subset is closed if and only if it is closed under limits of sequences (in the original topology). Spaces which satisfy the first axiom of countability are always sequential (indeed, they are Fréchet–Urysohn spaces), and the sequential spaces form the smallest coreflective subcategory containing all first-countable spaces. For this reason, many topological results which are traditionally proved for first-countable spaces can readily be extended to sequential spaces.
References
| [a1] | R. Engelking, "General topology" , Heldermann (1989) |
How to Cite This Entry:
Sequential space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sequential_space&oldid=33099
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article