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From Wikipedia:

The first requirement is that X is in T. If T is a subset of X, how can X be in T? Isn't that a paradox, sort of like Russell's paradox? How can a set be contained within a subset of itself? Am I interpreting the axiom wrong?Formal definition

Formally, a topological space is a set X together with a collection T of subsets of X (i.e., T is a subset of the power set 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 pair of sets in T is also in T.

Also, in reading about spaces in general, it seems like the general concept of space, metric spaces, euclidean space, etc .. are geometrical concepts. Objects like sets, groups, rings and fields are more algebraic in nature. However, the definition of a topological space seems to be more of an algebraic rather than a geometrical concept. Is that correct?