Definition:
$X$ is a sequential space if, whenever $A\subset X$ and $A$ is not closed, there is a sequence $\{a_n:n∈ω\}⊂A$ such that $a_n→y$ for some $y\in A^c$.
Is there any example to show that a sequential space need not be a Fréchet space?
Asked
Active
Viewed 511 times
2
Martin Sleziak
- 53,687
habib
- 23
-
what you mean by $Ac$? – nanthini Aug 01 '13 at 07:26
-
@nanthini: It was in fact $A^c$, the complement. – Seirios Aug 01 '13 at 07:37
-
1Do you mean an example of a space which is sequential, but not Frechet? From the current version it seems that you want an example showing that no space can be both sequential and Frechet. – SBF Aug 01 '13 at 07:39
-
1Are you sure you wanted to link to the Wikipedia article about Fréchet space and not about Fréchet-Urysohn space? The latter class of spaces is often called Fréchet spaces, too; but the link you provided is about kind of topological vector spaces. – Martin Sleziak Aug 01 '13 at 07:50
-
Very similar question (with somewhat confusing title - at this moment) is here. – Martin Sleziak Aug 01 '13 at 07:53
-
@seirios okay thank you – nanthini Aug 01 '13 at 08:52
1 Answers
4
Yes. The standard example is the Arens space, which is fully discussed in this post to Dan Ma’s Topology Blog. In fact, every sequential space that is not Fréchet contains a copy of the Arens space.
Brian M. Scott
- 616,228