# Tagged Questions

Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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### Uniformly Cauchy sequence of functions

I am trying to show the following: For each $n \in \mathbb N$, let $f_n:X \to Y$, where $(Y,d)$ is a complete metric. Suppose that for every $\epsilon>0$, there exists $n_0 \in \mathbb N$ such ...
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### Showing $f(a) \in V$ but $f(x_n) \notin V$ for every $n$

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces and let $f: X \rightarrow Y$ be a function. Let $a \in X$ and suppose $f$ is not continuous in $a$. Prove that there exists an open subset $V$ in $Y$ ...
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### Is this metric space normable?

Given the function $\rho:\mathbb{R}^n\to\mathbb{R}_+$ defined by $$\rho(x)=\log\left(\frac{\sum\left|x_i\right| e^{\left|x_i\right|}}{W\left(\sum\left|x_i\right| e^{\left|x_i\right|}\right)}\right)$$ ...
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### Urysohn Metrization Theorem contradiction (uniform topology homeomorphic to product topology)?

The theorem states that if $F$ is regular and has a countable basis, then it is metrizable. In Munkres' proof of this theorem, he gives a function (homeomorphism) $F:X \rightarrow [0,1]^\omega$ that ...