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gmvh
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Suppose that for all $n$ natural numbers, $d_{n}$ is a pseudometric on set $X $. Define $d=\sum_{n=1}^{\infty }a_{n}\frac{d_{n}}{1+d_{n}}$, where $\left( a_{n}\right) $ is a seqeuncesequence of positive numbers such that $\sum_{n\in %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion }a_{n}<\infty $.

It is claimed that the uniformity $\mathcal{U}_{d}$ generated by pseudometric $d$ is the same the uniformity $\mathcal{U}_{P}$ generated by pseudometrics $P=\left\{ d_{n}:n\in %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion \right\} $, that iseis $\mathcal{U}_{d}=\mathcal{U}_{P}$.

At page 237 in Topology Book by W. W. Fairchild, C. I. Tulcea,

https://archive.org/details/topology0000fair/page/236/mode/2up

Suppose that for all $n$ natural numbers, $d_{n}$ is a pseudometric on set $X $. Define $d=\sum_{n=1}^{\infty }a_{n}\frac{d_{n}}{1+d_{n}}$, where $\left( a_{n}\right) $ is a seqeunce of positive numbers such that $\sum_{n\in %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion }a_{n}<\infty $.

It is claimed that the uniformity $\mathcal{U}_{d}$ generated by pseudometric $d$ is the same the uniformity $\mathcal{U}_{P}$ generated by pseudometrics $P=\left\{ d_{n}:n\in %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion \right\} $, that ise $\mathcal{U}_{d}=\mathcal{U}_{P}$.

At page 237 in Topology Book by W. W. Fairchild, C. I. Tulcea,

https://archive.org/details/topology0000fair/page/236/mode/2up

Suppose that for all $n$ natural numbers, $d_{n}$ is a pseudometric on set $X $. Define $d=\sum_{n=1}^{\infty }a_{n}\frac{d_{n}}{1+d_{n}}$, where $\left( a_{n}\right) $ is a sequence of positive numbers such that $\sum_{n\in %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion }a_{n}<\infty $.

It is claimed that the uniformity $\mathcal{U}_{d}$ generated by pseudometric $d$ is the same the uniformity $\mathcal{U}_{P}$ generated by pseudometrics $P=\left\{ d_{n}:n\in %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion \right\} $, that is $\mathcal{U}_{d}=\mathcal{U}_{P}$.

At page 237 in Topology Book by W. W. Fairchild, C. I. Tulcea,

https://archive.org/details/topology0000fair/page/236/mode/2up

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Mehmet Onat
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A question about uniformities generated by pseudometrics

Suppose that for all $n$ natural numbers, $d_{n}$ is a pseudometric on set $X $. Define $d=\sum_{n=1}^{\infty }a_{n}\frac{d_{n}}{1+d_{n}}$, where $\left( a_{n}\right) $ is a seqeunce of positive numbers such that $\sum_{n\in %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion }a_{n}<\infty $.

It is claimed that the uniformity $\mathcal{U}_{d}$ generated by pseudometric $d$ is the same the uniformity $\mathcal{U}_{P}$ generated by pseudometrics $P=\left\{ d_{n}:n\in %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion \right\} $, that ise $\mathcal{U}_{d}=\mathcal{U}_{P}$.

At page 237 in Topology Book by W. W. Fairchild, C. I. Tulcea,

https://archive.org/details/topology0000fair/page/236/mode/2up