Need help with this proof

Cratylus

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Prove using the def. of closed set that S=[0,\(\displaystyle \omega\)) is not a closed subset of X=[0,\(\displaystyle \omega\)] with order topology(prob 4.a,pg 68)


4.1 Definition Let X. be a topological space and let x\(\displaystyle \in\) X . A set N\(\displaystyle \in\) X is a
neighborhood of x if there an open set U\(\displaystyle \in\) X s.t x\(\displaystyle \in\) U\(\displaystyle \subset\) N

Attempted Proof(via contradiction)
Let (X,T) be a topology.S being the set of countable ordinals, it is Hausdorff.(They many open sets contained in
S)
This implies that if x \(\displaystyle \ne\)y, there are two open sets U and V, s.t x\(\displaystyle \in\) U ,y\(\displaystyle \in\) V
with U \(\displaystyle \cap\) V=0. But X\S is open and x\(\displaystyle \in\) U\(\displaystyle \subset\) N\S
and y\(\displaystyle \in\)V\(\displaystyle \subset\) N\S in S. with U \(\displaystyle \cap\) V=0
But this is impossible . Thus S is not closed in X

I am using the book A First Course in Topology by Robert Conover
l can assume everythinh up to this point and info on ordinals.
Any help would be appreciated.
 

pka

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Prove using the def. of closed set that S=[0,\(\displaystyle \omega\)) is not a closed subset of X=[0,\(\displaystyle \omega\)] with order topology(prob 4.a,pg 68)
4.1 Definition
Let X. be a topological space and let x\(\displaystyle \in\) X . A set N\(\displaystyle \in\) X is a
neighborhood of x if there an open set U\(\displaystyle \in\) X s.t x\(\displaystyle \in\) U\(\displaystyle \subset\) N
Please tell what topology textbook from which that comes. Your terminology and notation us not standard at all.
A topological space is simply a set, any set,\(\mathcal{X}\). That set contains collection of subsets, \(\mathscr{T}\) having the properties:
\(\bf{[\mathcal{O}_1]}\quad\)The sets \(\mathcal{X}~\&~\emptyset\) both belong to \(\mathscr{T}\).
\(\bf{[\mathcal{O}_2]}\quad\)The union of any collection of sets from \(\mathscr{T}\) belongs to \(\mathscr{T}\). \(\mathscr{T}\) is closed under arbitrary union.
\(\bf{[\mathcal{O}_3]}\quad\)The intersection of two sets from \(\mathscr{T}\) belongs to \(\mathscr{T}\). Closed under finite intersection.
The sets in \(\mathscr{T}\) are the open sets in the topology on \(\mathcal{X}\).
Now does that correspond with what you know as a topological space?
B.T.W You mention a set \(S=[0,\omega]\) without defining \(\omega\). What is it?
 

Cratylus

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Please tell what topology textbook from which that comes. Your terminology and notation us not standard at all.
A topological space is simply a set, any set,\(\mathcal{X}\). That set contains collection of subsets, \(\mathscr{T}\) having the properties:
\(\bf{[\mathcal{O}_1]}\quad\)The sets \(\mathcal{X}~\&~\emptyset\) both belong to \(\mathscr{T}\).
\(\bf{[\mathcal{O}_2]}\quad\)The union of any collection of sets from \(\mathscr{T}\) belongs to \(\mathscr{T}\). \(\mathscr{T}\) is closed under arbitrary union.
\(\bf{[\mathcal{O}_3]}\quad\)The intersection of two sets from \(\mathscr{T}\) belongs to \(\mathscr{T}\). Closed under finite intersection.
The sets in \(\mathscr{T}\) are the open sets in the topology on \(\mathcal{X}\).
Now does that correspond with what you know as a topological space?
B.T.W You mention a set \(S=[0,\omega]\) without defining \(\omega\). What is it?
A First. Course in Topology Robert Conover
l know the definition of a topology.
\(\displaystyle \omega\)={0,1,2,…,n,…:n\(\displaystyle \in\)Z+}
s.t is short for such that
l thought these were standard notation
Typo in definition ,it should say A set N\(\displaystyle \subset\) X is a neighborhood…
Me bad
So l think you are inferring if l can give a counter example,that would suffice?
 
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pka

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A First. Course in Topology Robert Conover
l know the definition of a topology.
\(\displaystyle \omega\)={0,1,2,…,n,…:n\(\displaystyle \in\)Z+}
l thought these were standard notation
So l think you are inferring if l can give a counter example,that would suffice?
I must tell you that the name Robert Conover is not in the data base of PhD mathematicians. See Here.
This is a perfect example of my concern for you basic understanding. I doubt that you have any.
You say \(\omega={0,1,2,…,n,…:n\in Z^+}\) That notation is totally meaninglesss
If you really want to learn topology get a reparable textbook such as Principles of Topology, by Fred Croom SEE HERE
For less than $20u.s. you will save your self a lot of worry.
 

Cratylus

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I must tell you that the name Robert Conover is not in the data base of PhD mathematicians. See Here.
This is a perfect example of my concern for you basic understanding. I doubt that you have any.
You say \(\omega={0,1,2,…,n,…:n\in Z^+}\) That notation is totally meaninglesss
If you really want to learn topology get a reparable textbook such as Principles of Topology, by Fred Croom SEE HERE
For less than $20u.s. you will save your self a lot of worry.
Here is the whole quote “
The ordinal number \(\displaystyle \omega\) is the smallest infinite(transfinite) ordinal number
It is the smallest ordinal number which is larger than every finite ordinal number,and in
canonical form \(\displaystyle \omega\)={0,1,2,3,…,n,…:n\(\displaystyle \in\) Z+}
Thus the smallest infinite ordinal number is(in the canonical form) just the set in the usual way.
pg 26
More info on \(\displaystyle \omega\)
l am surprised you never heard of it. When you quoted the definition of a topology,l could
only use it based on open sets.
The definition of a closed set l could use was Let X be a topological space. A subset F
\(\displaystyle \subset\) X is closed if it’s complement ( X-F ) is open in X

l do have some knowledge of topology.Croom mentions the same definition of a finite ordinal
in his text as Conover. His stuff on metric spaces is the same too...

l have Bert Mendelson lntro to Topology
 
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