# Meaning of Space?

## Main Question or Discussion Point

What is the meaning of 'Space' (in the context of vector spaces, Banach spaces, etc)? Is space just another name for a set?

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It depends on the context... But in general, a space is a set (or class or collection of mathematical objects) with some special properties (e.g. it is equipped with certain operations which in turn satisfy certain requirements). In essense, it's the universe you're going to work in, hence the name.

So, for example, a Boolean Space $$\mathcal{B}$$ would be a set of two elements 0 and 1 equipped with two binary operations $$\lor$$ and $$\land$$ and an unary operation $$\lnot$$ such that the usual axioms of associativity, commutativity, distributivity, etc hold. Right?

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mathwonk