Right Ray Topology: Excluded Set Topology

[jjanks88]

Topology Excluded set

<< Original text of post restored by Mentors >>

Consider the topology T={u⊆ℝ : u=ℝ or u∩[0,3]=∅}, an “excluded set” topology on ℝ.

a.) Prove that (ℝ, T) is indeed a topological space
b.) Give an example of a closed subset (not ℝ) which contains 2.
c.) Give a different example of a closed subset which contains 5.
d.) Compute Cl({2}) and Cl({4})
e.) For J =[-1,1], compute Int(J), Ext(J), Bdy(J)
f.) Prove that (ℝ,T) is compact, or show it isn’t
g.) Is [2,4] a compact subset of this space? Is [4,6]?
h.) Prove that (ℝ, T) is connected, or show it isn’t.

I know this is a lot of problems but I wish the book had answers in the back of the book! Sorry, please help me. I need some insurance so that I can feel safe I am doing stuff right later on. Thanks!

 

[micromass]

Please provide an attempt at the solution.

 

[jjanks88]

Please provide an attempt at the solution.

Sorry...

a.)
The properties
The empty set and X are in τ.
The union of any collection of sets in τ is also in τ.
The intersection of any pair of sets in τ is also in τ

b.) [0,3]
c.)[4,5]
d.)[0,3] and emptyset
e.)Not sure
f.)Not sure
g.)??
h.)??

 

[micromass]

Sorry...

a.)
The properties
The empty set and X are in τ.
The union of any collection of sets in τ is also in τ.
The intersection of any pair of sets in τ is also in τ

Yes, and what did you try already to show these things? The first one should be pretty easy...

b.) [0,3]

Correct.

c.)[4,5]

Not correct, this is not a closed set.

d.)[0,3] and emptyset

[0,3] is correct. The empty set is not correct, since the closure of a nonempty set can NEVER be empty!

So, try to solve the first problem first, tell me where your stuck instead of simply telling me what you think the answer is. If you do that, then I'll know where to help...