Thread: 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 = {T_n :n ∊N} ∪ {∅} where T_n ={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.

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 = {T_n :n ∊N} ∪ {∅} where T_n ={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 look like?

What would the union of any collection of look like?

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?

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.