What is the definition of 'space'?

[quasar987]

Mathworld's definition: "The concept of a space is an extremely general and important mathematical construct. Members of the space obey certain addition properties."

It is quite vague. What would a rigourous definition be?

 

[HallsofIvy]

Notice that it says "a space" not just "space". There are a number of different kinds of "spaces" used in mathematics (vector space, topological space (in which members do not "obey certain addition properties"!), etc.).

If you want a definition of "space" (that in which we all live), then you should ask in a physics forum.

 

[dextercioby]

In the most abstract sense,u can't have a space without a set.So the notion of "set" is the trully elementary one.A space is a set whose elements & subsets have certain properties.U can't be too rigurous,really.

Daniel.

 

[jcsd]

There isn't really a rigourous definition, but ea space si basically a ste with some additonal structure defined on it.

So for exmaple a vector space has the addiotnal structure of an asosciated field, vector additon and scalar mulpilcation; a metric space has the additonal structure of a metric function; a topological space as the additonal structure of a topology and so on.

 

[snoble]

I bet there's got to be a category theorist out who can take these general notions and come up with a general rigorous definition. Like a space is a collection of sets, an operation between the sets and a set of axioms that must be satisfied. ie one of the sets in the collection would be the elements of the space, another set could be a topology on the space and the operation between those two sets would be inclusion.

I'm not saying this would be the categorist's definition but I bet there is one. A categorist just can't help but to generalize these sorts of things.