Question is said in title.Suppose X is a separable topological space and S is a dense subspace of X,to prove S is separable or give a counter-example,and we may add what conditions to make this statement true?
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1What is a divisible topological space? – Luiz Cordeiro Sep 19 '16 at 05:23
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@LuizCordeiro sorry,it's separable,i've got it wrong – user360777 Sep 19 '16 at 05:27
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You can add the condition that $S$ is countable :) – Exit path Sep 19 '16 at 05:35
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@LuizCordeiro but X{a} is not dense in X here – user360777 Sep 19 '16 at 05:47
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http://math.stackexchange.com/questions/892041/is-every-dense-subspace-of-a-separable-space-separable – Luiz Cordeiro Sep 19 '16 at 05:58
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@LuizCordeiro why in the first counter-example the sigma product is dense in pi product – user360777 Sep 19 '16 at 06:49
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This is from the definition of the product topology: If $(x_i)\in\prod_I[0,1]$, a basic open neighbourhood of $(x_i)$ has the form $\left{(y_i):\operatorname{max}_{i\in F}|x_i-y_i|<\epsilon\right}$, where $F\subseteq I$ is finite and $\epsilon>0$. Then setting $y_i=x_i$ for $i\in F$ and $0$ otherwise, we have $(y_i)$ in this neighbourhood. – Luiz Cordeiro Sep 19 '16 at 06:52
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@LuizCordeiro thanks! – user360777 Sep 19 '16 at 07:13