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Here are two important generalizations of the notion of topological space:

1. C-algebras.
   Given a space X, one considers the set of continuous functions from X to ℂ. One then looks at all the properties of this set (it's an algebra, it's a Banach space, ...). One relaxes a bit the conditions (allow the multiplication to be non-commutative): there you get the notion of a C-algebra.

2. Stacks.
   Given a space X, one considers the collection of all maps Y→ X, where Y is an arbitrary space. We then look at at their properties (they form a category over Top¹, there exists a notion of precomposition with a map Y'→Y, they behave well w.r.t open covers, ...). Relax some conditions, and you'll get the notion of a stack.

By focusing on out of a space X, you get the notion of C*-algebra.
By focusing on maps into a space X, you get the notion of stack.

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¹Here, *Top* refers to the category of topological spaces. Stacks are encountered more often in algebraic geometry. In that case, one uses the category of schemes in place of *Top*.