# Proof of topological space

• Mar 25th 2012, 08:35 PM
monster
Proof of topological space
I'm new to topology and have just started looking at whether spaces are topologies or not, I've come across a question where i have to show a space is a topology on N and am not sure how to approach it,

T = {Tn :n ∊N} ∪ {∅} where Tn ={m ∊N:m ≥ n}

I am supposed to prove T is a topology on N (natural numbers).
Clearly condition T1 is satisfied as T has the empty set and N, but i'm not sure how to approach showing T2 and T3 are satisfied any help would be great, cheers.
• Mar 26th 2012, 02:39 AM
Plato
Re: Proof of topological space
Quote:

Originally Posted by monster
I'm new to topology and have just started looking at whether spaces are topologies or not, I've come across a question where i have to show a space is a topology on N and am not sure how to approach it,
T = {Tn :n ∊N} ∪ {∅} where Tn ={m ∊N:m ≥ n}
I am supposed to prove T is a topology on N (natural numbers).
Clearly condition T1 is satisfied as T has the empty set and N, but i'm not sure how to approach showing T2 and T3 are satisfied any help would be great, cheers.

What would the intersection of any two \$\displaystyle T_n\$ look like?

What would the union of any collection of \$\displaystyle T_n\$ look like?
• Mar 26th 2012, 02:51 AM
monster
Re: Proof of topological space
may have suffered from not reading the question properly, is it that Union of say T1 and T2 = T2 which is contained in T hence condition 2 satisfied?
• Mar 26th 2012, 03:02 AM
Plato
Re: Proof of topological space
Quote:

Originally Posted by monster
may have suffered from not reading the question properly, is it that Union of say T1 and T2 = T2 which is contained in T hence condition 2 satisfied?

You must show closure of arbitrary unions and finite intersections.