# 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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### Proof that the nowhere differentiable functions are dense in $C_b(\mathbb R)$.

I tried to make a proof, where I use a Weierstrass function. I was surprised at how easy it was, and thus a little doubtful as to the correctness of the proof. I've looked it over, and didn't find any ...
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### Proof idea: Let $(X,d)$ be a metric space, and $\rho$ be bounded metric, show that they will generate the same topology

Let $(X,d)$ be a metric space, $d$ generates the metric topology $\mathcal{T}$ via metric ball $B_\epsilon(x)$. Show that bounded metrics: $\rho_1(x,y) = \dfrac{d(x,y)}{1+d(x,y)}$ with ...
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### Show that $d_1=\min(d(x,y),2)$ is a metric space [duplicate]

Show that $d_1=\min(d(x,y),2)$ is a metric space if it is given that $d(x,y)$ is a metric space. I am stuck at the triangle inequality part, to show that $d_1(x,z)\leqslant d_1(x,y)+d_1(y,z)$ i.e ...
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### Weak measurability of a set-valued map

Suppose that $A$ and $B$ are compact metric spaces. Let $f:A\times B\to B$ be a Borel measurable map (in the sense that for every Borel set $S\subseteq B$, $f^{-1}(S)$ belongs to the $\sigma$-algebra ...