Similarity and Difference between Separable Space and Separated space?

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Does separability and/or second countability implies $T_2$ or higher axiom sets?
My intuition is "no". Even $T_0$ space can be separability and/or second countability?
general-topology separation-axioms separable-spaces second-countable
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Does separability and/or second countability implies $T_2$ or higher axiom sets?
My intuition is "no". Even $T_0$ space can be separability and/or second countability?
general-topology separation-axioms separable-spaces second-countable
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up vote
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up vote
0
down vote
favorite
Does separability and/or second countability implies $T_2$ or higher axiom sets?
My intuition is "no". Even $T_0$ space can be separability and/or second countability?
general-topology separation-axioms separable-spaces second-countable
Does separability and/or second countability implies $T_2$ or higher axiom sets?
My intuition is "no". Even $T_0$ space can be separability and/or second countability?
general-topology separation-axioms separable-spaces second-countable
general-topology separation-axioms separable-spaces second-countable
edited yesterday
Asaf Karagila♦
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No. Consider $X = {a,b}$ just two elements and $mathcal{T} = {emptyset, X}$. This is separable, second countable, and not even $T_0$.
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1 Answer
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active
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up vote
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No. Consider $X = {a,b}$ just two elements and $mathcal{T} = {emptyset, X}$. This is separable, second countable, and not even $T_0$.
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up vote
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No. Consider $X = {a,b}$ just two elements and $mathcal{T} = {emptyset, X}$. This is separable, second countable, and not even $T_0$.
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up vote
1
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up vote
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No. Consider $X = {a,b}$ just two elements and $mathcal{T} = {emptyset, X}$. This is separable, second countable, and not even $T_0$.
No. Consider $X = {a,b}$ just two elements and $mathcal{T} = {emptyset, X}$. This is separable, second countable, and not even $T_0$.
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mathworker21
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