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I recently read that given a topology , a set is said to be open if it is in $T$. This is quite different what read about open sets in Euclidean space. An article in Wikipedia states that

if about one point in a topological space there exists an open set not containing another (distinct) point, the two points are referred to as topologically distinguishable. In this manner, one may speak of whether two subsets of a topological space are "near" without concretely defining a metric on the topological space. Therefore, topological spaces may be seen as a generalization of metric spaces.

Suppose $X = \{ a, b, c\}$ and let $T = \{ \emptyset,\{a \}, X \} $. $T$ is topology on $X$ and $\{a\} \in T$ is an open set. How does it relate to the above statement?

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The open set $\{a\}$ contains only $a$, so the quoted statement means that $a$ is topologically distinguishable from both $b$ and $c$. The only open set containing $b$ is $X$. The only open set containing $c$ is also $X$, so the quote would suggest that $b$ and $c$ are topologically indistinguishable. In this sense we would say that $b$ and $c$ are "near" to each other, but $a$ and $b$ are not. Be careful, though. Nearness is not a rigorously defined topological notion, and some topological spaces do not admit a metric to quantify nearness. The topology you have given, for instance, does not admit a metric.

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