# Need help with this proof

#### Cratylus

##### New member
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

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

##### Elite Member
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

##### New member
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…
So l think you are inferring if l can give a counter example,that would suffice?

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#### pka

##### Elite Member
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 ##### New member 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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#### Cratylus

##### New member

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,$$\displaystyle \Omega$$ ] and X=[0,$$\displaystyle \Omega$$ ] and $$\displaystyle \Omega$$ is the Initial ordinal of card($$\displaystyle \aleph_1$$ ). I have a bad habit of miscopying.