Linked Questions

1
vote
2answers
122 views

Proving Non-metrizable spaces [duplicate]

Maybe this question seems easy, but is there a strategy or a property to look for when demostrating a space is non-metrizable? Because a part from the fact that I can't find a metric generating the ...
10
votes
2answers
4k views

Weak topology on an infinite-dimensional normed vector space is not metrizable

I've been pondering over this problem for a while now, but I can't come up with a proof or even a useful approach... Let $X$ be am infinite-dimensional normed vector space over $\mathbb{K}$ (that is ...
4
votes
5answers
4k views

Non-Metrizable Topological Spaces

What are some motivations/examples of useful non-metrizable topological spaces? I am trying to get a feel for what parts of math have topologies appear naturally, but not induced by a metric space. ...
10
votes
3answers
856 views

Topological spaces vs. metric spaces

Are there any "realistic" examples of topological spaces that are not metric spaces. You are free to invent your own definition of "realistic". But, at a minimum, a realistic example is one that ...
3
votes
1answer
2k views

Two questions on the Zariski topology on $\mathbb{R}$

Consider the Zariski topology on the set $\mathbb{R}$. 1) Is the set $(0,1)$ compact in this topology? I said that it was because under the Zariski topology it was closed as there are infinitely ...
5
votes
2answers
2k views

The weak$^*$ topology on $X^*$ is not first countable if $X$ has uncountable dimension.

I learnt without proof that if $X$ is a normed space of uncountable dimension, then the weak* topology on $X^*$ is not first countable. Can anyone point out how I should go about proving it? I tried ...
10
votes
1answer
2k views

$C(X)$ with the pointwise convergence topology is not metrizable

I need to show that if $X$ is an uncountable Tychonoff space, then $C(X)$ is not metrizable. All I've been able to show so far is that that $F(X)$, the space of all functions with pointwise topology, ...
5
votes
1answer
729 views

Weak topology is not metrizable: what's wrong with this proof?

Let $(X,\|\cdot\|)$ be an infinite-dimensional normed vector space. Suppose that the weak topology of $X$ is metrizable by a metric $d$. Denote by $B^d(x,r)$ the open balls with respect to $d$; they ...
2
votes
2answers
678 views

The space of distributions endowed with the topology of uniform convergence on bounded sets is not Fréchet.

I found a state, that the space of distributions on (here:) $\mathbb{R}^{n}$, which is equipped with the topology of uniformly convergence on bounded subsets is not a Fréchet space. As far as i can ...
1
vote
2answers
320 views

What is the diagonal principle?

I'm guessing I'm supposed to show that there exists a subsequence that converges in probability quickly (hypothesis of $(c)$). What is the diagonal principle? Is that related to Cantor's ...
2
votes
1answer
272 views

Topological Vector Space not induced by Metric

Can anyone give me an example of a Topological Vector Space that is not metrizable? I know that the neighborhood base of $0$ needs to be incountable, and all I can construct then is no topological ...