Consider a set X.
Is X a topological space?
Not yet. To be a topological space, X needs to have a topology defined on it.
What’s a topology?
A topology T on X is a collection of subsets of X that follows certain conditions.
I’ll ask about those conditions in a minute. First, I just want to clarify: is T a set?
Yes.
And the elements of T are each subsets of X?
Yes, that’s correct.
So what are these special properties?
To be a topology, T must contain the empty set.
I thought you said the elements of T had to be subsets of X. Is the empty set a subset of X?
Yes. The empty set is a subset of any set.
Ah, okay. What’s another condition?
To be a topology, T must also contain X.
And X is a subset of X I guess?
Yes, that’s right.
Are they the only two conditions?
No, there are two more.
This doesn’t seem too bad so far.
Glad to hear it.
What’s the third condition?
The intersection of any two sets in T must also be in T.
That sounds interesting. And the fourth condition?
The union of any two sets in T must also be in T.
You know, I thought it might be something like that. So, is that all there is to being a topology?
Yes. If:
then T is a topology on X.
So, if X = {1} then T = {Æ, {1}} is a topology on X?
Yes.
And if X = {1,2} then T = {Æ, {1}, {2}, {1,2}} is a topology on X?
Yes.
And if X = {1,2,3} then T = {Æ, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} is a topology on X?
Yes. Well done!
So, if T contains every subset of X including Æ and X itself, then T is a topology on X?
Yes. The set of all subsets of X is called the power set of X.
So the power set of X is always a topology on X.
Yes, that’s correct.
Can X have other topologies besides the power set?
Yes it can. In fact, {Æ, X} is always a topology on X.
Ah, yes. Because ÆÇX = Æ and ÆÈX = X, all the rules for a topology are met by {Æ, X} for any X.
Yes. Well spotted.
And {Æ, X} must be the smallest topology possible on X.
Yes. Because anything smaller won’t meet all the rules.
Now that I think about it, the power set must be the biggest topology possible for a set.
Quite right.
Can there be any topologies in between?
See if you can find one for X = {1,2}.
Hmm. Would T={Æ, {1}, {1,2}} be a topology on X?
Is every element of T a subset of X?
Yes. ÆÍX, {1}ÍX and {1,2}ÍX.
Does T contain the empty set?
Yes it does. ÆÎT.
Does T contain X?
Yes. X={1,2}ÎT.
Is the intersection of any two sets in T also in T?
Let me try them:
Is the union of any two sets in T also in T?
Let me try them:
All the conditions have been met.
So I've just proved that {Æ, {1}, {1,2}} is a topology on {1,2}!
Yes you have. Well done!
I have a question.
Sure.
Do the elements of T have a name?
Yes. They are called open sets.
So, defining a topology on X is defining which subsets of X are open sets?
Very perceptive of you. That's exactly what you are doing when you define a topology.
So, the rules about what makes a topology can be thought of as rules that open sets follow:
Spot on.
If there are things called "open sets", are there things called "closed sets"?
Yes there are. A closed set is any subset of X that isn't open.
In other words, any subset of X that isn't in T.
Correct.
So in the example of the topology T={Æ, {1}, {1,2}}, the only closed set is {2}.
Correct.