# Linked Questions

11 questions linked to/from Prove that some topology is not metrizable
2answers
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### 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 ...
2answers
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### 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 ...
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### 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. ...
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 ...
1answer
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### 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 ...
2answers
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### 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 ...
1answer
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### $C(X)$ 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, ...
1answer
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### 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 ...
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 ...
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 ...
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 ...