Proving T is a topology.

1. Sep 1, 2012

cragar

1. The problem statement, all variables and given/known data
Let X be a set and p is in X, show the collection T, consisting of the empty set and all the subsets of X containing p is a topology on X.

2. Relevant equations?
A topology T on X is a collection of subsets of X.
i) X is open
ii) the intersection of finitely many open sets is open
iii) the union of any collection of open sets is an open set.
3. The attempt at a solution
How do I know that X is open. Maybe open doesn't mean the same thing as an open set on the reals.
if X is open. And If I take an intersection of a finite number of sets in T i will get something in T so by definition It will produce an open subset in T. And all my intersections will contain P because P is in every set.
for iii) If I take a union of all the subsets in T I will produce a subset in T containing p. I could not produce something outside of T because i started with sets that were part of
T.

2. Sep 1, 2012

Bacle2

Does X contain p? This is the way of telling if X is open in your topology.

3. Sep 2, 2012

cragar

so because we know that p is in X , we know that it is open.
Is a topology just a way of defining properties on our weird set.

4. Sep 2, 2012

Bacle2

Well, after a lot of time , it was found that many of the properties that were

usually studied in topology could be defined in terms of open sets: continuity,

compactness,etc. And, yes, by assigning a topology to a set ,we gain the ability

to talk about topological concepts in the set. And with different topologies, we

end up with different topological properties.

5. Sep 2, 2012

HallsofIvy

Staff Emeritus
Are you sure you have copied this correctly? You seem to be defining a "topology T" but then don't mention "T" in the definition! But you do talk about "open sets" without defining them.

What is your definition of "open" set? Normally, in General Topology, we define a topology, T, of X as a collection of subsets of X satisfying:
1) X is in T
2) The empty set is in T
3) The union of any subcollection in the T
4) The intersection of any finite subcollection is in the T

We then define a set to be "open" if and only if it is in the collection.

Last edited: Sep 2, 2012
6. Sep 2, 2012

Bacle2

I thought we define a set U to be open if for every x in U there is an element Ux of T

as above, such that x is contained in Ux , and Ux is in T.

7. Sep 2, 2012

cragar

ok thanks for your help guys

8. Sep 2, 2012

vela

Staff Emeritus
What do you mean by Ux?

It's been awhile since I took topology, but I remember it the way HallsofIvy said. If a set is in T, it is, by definition, open.

9. Sep 2, 2012

SammyS

Staff Emeritus
Yes vela & Halls are correct. In fact the topology described in this problem is called the particular point topology (or included point topology).

10. Sep 2, 2012

SteveL27

The particular point topology is the answer to an interesting question.

We know that if K is a compact topological space, every continuous function from K to the reals is bounded.

Now, if X is a topological space such that every continuous function from X to the reals is bounded, must X be compact?

The answer is (surprisingly) no. The counterexample is any infinite set with the particular point topology. The proof is simple and amusing.

Last edited: Sep 2, 2012