Everyone has heard of so-called Euclidean space, the set of points in which we have practised geometry ever since coordinate systems and coordinates entered the mathematical world. With a few adjustments and some effort to model reality, three-dimensional space corresponds broadly to what we can perceive of the environment around us, while two-dimensional space is the plane on which we represent things, from the blackboard to the notebook page.
Space in all its forms
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Yet the abstract notion of mathematical space is very different from what intuition tells us. It reflects a desire to classify and map knowledge, but also, in the interests of universality, to bring similar ideas together. A space is a kind of "working environment" for mathematicians, obtained by equipping a set with a structure—a "toolbox"—drawn from algebra, analysis, or even geometry.
Standard arguments and calculations performed with a given structure—for example, over the real numbers—remain valid when the object under study changes. The same techniques can therefore be applied to sets of numbers, vectors, functions… or even sets of sets.
When a set E is equipped with an algebraic structure—operations + and ∙ satisfying certain conditions—it becomes a vector space, in which algebraic and geometric arguments can be made. If the same set is equipped with a topological structure, it becomes a topological space, taking us into analysis: limits, continuity of functions, convergence of sequences, and so on.
