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## Main Question or Discussion Point

A topological space is a set X together with T, a collection of subsets of X, satisfying the following axioms:

1.The empty set and X are in T.

2.The union of any collection of sets in T is also in T.

3.The intersection of any finite collection of sets in T is also in T.

The sets in T are the open sets

I can't see how it can become the open set definition in metric space.

Say,in 2-D Euclidean space,let X be a circle with all its interior points, then a trivial topology is T={{}, X}, it seems not to contradict to the axioms, but X is a closed set , can anybody help me clarify this? Thanks in advance.

1.The empty set and X are in T.

2.The union of any collection of sets in T is also in T.

3.The intersection of any finite collection of sets in T is also in T.

The sets in T are the open sets

I can't see how it can become the open set definition in metric space.

Say,in 2-D Euclidean space,let X be a circle with all its interior points, then a trivial topology is T={{}, X}, it seems not to contradict to the axioms, but X is a closed set , can anybody help me clarify this? Thanks in advance.