Definition of Topology: What is a Topological Space?

[Greg Bernhardt]

Definition/Summary

A Topological Space can be defined as a non empty set X along with a class of sets, called a topology on X which is closed under (1) arbitrary unions (2) finite intersections. It can be assumed that this will always include X and p (the empty set), but understanding the reason for this can be quite difficult.


Equations

Let

X be the universe
P be the empty collection
Q be the collection of compliments of P
p the empty set
U = Union
I = Intersection
C = Compliment

Briefly, p is included as the union of the empty collection and X is included as the intersection of the empty collection. It's the latter of these that causes conceptual difficulties. How can the intersection of a collection with nothing in it be the whole universe?

If we first accept that as a simple accumulation of elements it's easy to grasp that U(P) = p. Now we can use De Morgan's laws to calculate I(P). Normally when we deal with these laws we have a non empty collection of sets. So, the collection of compliments is also a non empty collection. In our case here though, there are no sets to have compliments of, so there are no compliments, so Q = P and thus U(Q) = U(P) = p. Now, by De Morgan:

The compliment of the union of a collection = The intersection of the collection of compliments.

So, C(U(P)) = I(Q)

But, Q = P, so I(Q) = I(P)

Also, C(U(P)) = C(p) = X

Thus I(P) = X

Extended explanation

This is basically the key point that makes sense out of the logic of this situation. If you accept that U(P) = p and you get that Q = P, the result necessarily follows.

Another way of thinking about this involves returning to the ideas of basic set logic. Given a universe X, then any sub set A can be seen as a condition on the elements of X. EG: Consider the universe of all people visiting a fair ground. P1 = the collection containing C1 = {people taller than 130cm}, C2 = {people older than 13}, ...Cn = {people with no heart condition} ... etc. These are the conditions on a particularly dangerous fair ground ride. The intersection of this collection of sets/conditions is I(P1) = {people allowed to go on ride 1}, that is, only those people who satisfy all of the conditions. Now consider another ride, that's very safe and has no conditions of entry. This collection of conditions, P2, is empty and the intersection of this collection is I(P2) = {people allowed to go on ride 2} = X, that is, everyone.

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[fresh_42]

Here's a nice read about it, too: https://ncatlab.org/nlab/show/topology