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I know that its a topology for Ex 2.6, but I dont know how to explain it. I know that T is a topology on a collection of subsets of X if it satisfies the axioms: 1. null set belong to T, and X belong to T. 2. If say U is a sub-collection of T, then the union of U should belong to T, and the same concept or theory applies for a finite sub collection of T.

Can someone help me on this, any help will be greatly appreciated. Thanks!!

Math_Is_Fun
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  • For 2.8 maybe read this first to get ideas? or this? – Henno Brandsma Jan 27 '20 at 23:28
  • Ex.2.6 very easy to explain, just identify the three conditions, first, $\varnothing,X\in\mathcal{T}$. Then, try all possible union and intersection of elements of $\mathcal{T}$ and you'll know that the condition 2 and 3 are both satisfied. And you're done. – Kevin.S Jan 28 '20 at 13:41

1 Answers1

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Solution for 2.6;

  1. $\phi$ and $X$ are in $\tau$

  2. Union of any members of $\tau$ is again in $\tau.$\ (check!)

  3. Intersection of any members of $\tau$ is again in $\tau.$ (check!)

Hence $\tau $ is a topology on $X.$ Follow the same arguments for other two problems as well!

user159888
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