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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.

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 = {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 $\displaystyle T_n$ look like?

What would the union of any collection of $\displaystyle T_n$ look like?

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?

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.