I'm a noob starting out studying differential geometry and topology. Really probably somewhere in the multivariate calculus level, but I've been trying to understand the plethora of terminology I'm encountering with this higher math. I've been reading a lot on Wikipedia.org and PlanetMath.org, (two excellent sites by the way)and I've got a question about topological spaces. 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? 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?