Why is (X; T ) not a topological space?

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math25

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Hi, can someone please help me to understand why this is not a topological space?

Let X = R and let T = {empty set} U {R} U {[a , infinity): a is in R } Why is (X; T ) not a topological space?

thanks
 
math25 said:
Hi, can someone please help me to understand why this is not a topological space?
Let X = R and let T = {empty set} U {R} U {[a , infinity): a is in R } Why is (X; T ) not a topological space?
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Is the collection \(\displaystyle \mathcal{T}\) closed under arbitrary union?
 
its not closed.

So T has to be closed under arbitrary union in order to be a topological space?
 
math25 said:
its not closed. So T has to be closed under arbitrary union in order to be a topological space?
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What does it mean to say \(\displaystyle \mathcal{T}\) is a topology on \(\displaystyle \mathcal{X}~?\)
 
I got it , thanks.

A topology on X is a set T of subsets of X which
satisfies few properties one of them being that whenever T contains an arbitrary collection {U } of sets, it also contains their
union
 
Whoah just curious, how is T not closed under arbitrary union? I tried picking some arbitrary sets like [5, infinity), [6, infinity) and their union seems to be in T. What am I doing wrong?
 
thedarjeeling said:
Whoah just curious, how is T not closed under arbitrary union? I tried picking some arbitrary sets like [5, infinity), [6, infinity) and their union seems to be in T. What am I doing wrong?
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What is \(\displaystyle \bigcup\limits_{n \in {\mathbb{Z}^ + }} {\left[ {\frac{1}{n},\infty } \right)} =~? \)
 
Wait I'm not so sure I actually understand.

Recall the Sorgenfrey topology, which is the union of intervals [a,b), and is pretty much what the original question is asking.

My teacher said that (0,1) actually IS an open set in the topology, because (0,1) = infinite union (1/n, 1).

Obviously it wouldn't be called the Sorgenfrey topology if it weren't a topology.

So my question is, isn't (0, infinity) actually in the topology since it is the result of an infinite arbitrary union? So I'm not sure I understand the proposed counterexample that there exists an arbitrary union which results in something not in the topology.
 
I think you are confusing the topology generated by the set T and the actual set given.
 
Maybe I'm completely off track, but it looks like the Sorgenfrey topology to me, because it looks to me like the union (look at the original problem, lots of union symbols) of sets of the form [a, b), but in this case b happens to be infinity. Maybe I am looking at it wrong, and I must be, because otherwise there's a contradiction somewhere. So what am I doing wrong here?
 
thedarjeeling said:
Maybe I'm completely off track, but it looks like the Sorgenfrey topology to me, because it looks to me like the union (look at the original problem, lots of union symbols) of sets of the form [a, b), but in this case b happens to be infinity. Maybe I am looking at it wrong, and I must be, because otherwise there's a contradiction somewhere. So what am I doing wrong here?
Click to expand...

Maybe your professor did not adequately get across what the Sorgenfrey Topology is. To be honest, I had to google it. The first few lines of the Wikipedia page should convince you that the set T in this question is different from the topology you are asking about.
 
thedarjeeling said:
it looks like the Sorgenfrey topology to me, because it looks to me like the union (look at the original problem, lots of union symbols) of sets of the form [a, b), but in this case b happens to be infinity. Maybe I am looking at it wrong, and I must be, because otherwise there's a contradiction somewhere. So what am I doing wrong
Click to expand...
.
First of all, you are describing what is known as the Sorgenfrey Line.
It is generated by a basis \(\displaystyle \mathcal{B}=\{[a,b):~a,b\in\mathbb{R}\}\)
The collection \(\displaystyle \mathcal{B}\) generates the Sorgenfrey Line.
In the OP, you were asked if a similar collection was itself a topology not a basis.
It is not closed under arbitrary union. Therefore, the collection is not a topology.
 
Ok, so what does it mean to generate a topology? How does a collection of sets which is not a topology suddenly generate a topology?
 
Well I'm still a little thick, I actually read that before you posted it, but I still didn't understand it.

I *think* the key is that the topology is a union of the sets generated by the basis, but on the other hand, isn't the basis already a union of sets of that form? The OP directly has the union symbol in front of sets of that form, so doesn't that already qualify it to be a union of sets of that form?
 
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