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ABB
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A particular example of a topological vector spacespaces

I am looking for a topological vector space $(X,\tau)$ enjoying the following conditions:

1- $(X,\tau)$ is not locally convex.

2- There exists a metric $d$ on $X$ and a sequence $\{X_n\}$ of subsets of $X$ such that $X=\cup X_n$ and the topology $\tau$ on $X_n$ is relatively second countable and metrizable$d$-metrizable for every $n$.

A particular example of a topological vector space

I am looking for a topological vector space $(X,\tau)$ enjoying the following conditions:

1- $(X,\tau)$ is not locally convex.

2- There exists a metric $d$ on $X$ and a sequence $\{X_n\}$ of subsets of $X$ such that the topology $\tau$ on $X_n$ is relatively second countable and metrizable for every $n$.

A particular example of topological vector spaces

I am looking for a topological vector space $(X,\tau)$ enjoying the following conditions:

1- $(X,\tau)$ is not locally convex.

2- There exists a metric $d$ on $X$ and a sequence $\{X_n\}$ of subsets of $X$ such that $X=\cup X_n$ and the topology $\tau$ on $X_n$ is relatively second countable and $d$-metrizable for every $n$.

Source Link
ABB
  • 4.1k
  • 1
  • 11
  • 19

A particular example of a topological vector space

I am looking for a topological vector space $(X,\tau)$ enjoying the following conditions:

1- $(X,\tau)$ is not locally convex.

2- There exists a metric $d$ on $X$ and a sequence $\{X_n\}$ of subsets of $X$ such that the topology $\tau$ on $X_n$ is relatively second countable and metrizable for every $n$.