Space for mathematicians is an interesting set. Or we can say a set with structure. Trying to get used to this concept, I am listing a few spaces, with some explanation, here. By the way, members of such sets are usually called points.

  • Vector Space: members of such a space can be scaled and added. Here the definition is dependent on properties of the members.
  • Topological Space: in this space points have neighbourhoods. Or, the topological space is a set on which a collection of open sets (closed under unions and finite intersections) is defined.
  • State Space: This means different things to computer scientists, physicists, and control engineers. For control engineers, state space is the space in which members are tuples of state variables.
  • Metric Space: a metric or distance is defined between members of such a space.
  • Euclidean Space: for most of us, this is the space acquired by age eighteen.

Of course, there are many more spaces. I mean, if you are so inclined, you can look up Hilbert Space, Minkowski Space, Anti de Sitter Space, etc. Just remember, a space is a set you can talk about.