Need help with this proof

[Cratylus]

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


4.1 Definition Let X. be a topological space and let x[MATH]\in[/MATH] X . A set N[MATH]\in[/MATH] X is a
neighborhood of x if there an open set U[MATH]\in[/MATH] X s.t x[MATH]\in[/MATH] U[MATH]\subset[/MATH] 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 [MATH]\ne[/MATH]y, there are two open sets U and V, s.t x[MATH]\in[/MATH] U ,y[MATH]\in[/MATH] V
with U [MATH]\cap[/MATH] V=0. But X\S is open and x[MATH]\in[/MATH] U[MATH]\subset[/MATH] N\S
and y[MATH]\in[/MATH]V[MATH]\subset[/MATH] N\S in S. with U [MATH]\cap[/MATH] 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]

Prove using the def. of closed set that S=[0,[MATH]\omega[/MATH]) is not a closed subset of X=[0,[MATH]\omega[/MATH]] with order topology(prob 4.a,pg 68)
4.1 Definition Let X. be a topological space and let x[MATH]\in[/MATH] X . A set N[MATH]\in[/MATH] X is a
neighborhood of x if there an open set U[MATH]\in[/MATH] X s.t x[MATH]\in[/MATH] U[MATH]\subset[/MATH] 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]

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.
[MATH]\omega[/MATH]={0,1,2,…,n,…:n[MATH]\in[/MATH]Z+}
s.t is short for such that
l thought these were standard notation
Typo in definition ,it should say A set N[MATH]\subset[/MATH] X is a neighborhood…
Me bad
So l think you are inferring if l can give a counter example,that would suffice?

 

[pka]

A First. Course in Topology Robert Conover
l know the definition of a topology.
[MATH]\omega[/MATH]={0,1,2,…,n,…:n[MATH]\in[/MATH]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]

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 [MATH]\omega[/MATH] 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 [MATH]\omega[/MATH]={0,1,2,3,…,n,…:n[MATH]\in[/MATH] Z+}
Thus the smallest infinite ordinal number is(in the canonical form) just the set in the usual way.
pg 26
More info on [MATH]\omega[/MATH]

Ordinal number - Wikipedia

en.wikipedia.org

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
[MATH]\subset[/MATH] 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

 

[Cratylus]

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?

l made big mistake in copying theorem S=(0,[MATH]\Omega[/MATH] ] and X=[0,[MATH]\Omega[/MATH] ] and [MATH]\Omega[/MATH] is the Initial ordinal of card([MATH]\aleph_1[/MATH] ). I have a bad habit of miscopying.