This question is from "Introduction to Topology: Pure and Applied," by Colin Adams and Robert Franzosa.
Here's how the authors define a topology:
Let X be a set. A topology Ƭ on X is a collection of subsets of X, each called an open set, such that
(i) ∅ and X are open sets;
(ii) The intersection of finitely many open sets is an open set;
(iii) The union of any collection of open sets is an open set.
Here's where I am with this problem:
I strongly believe Ƭ is a topology, because the conditions seem to be met:
(i) Yes, ∅ and ℝ are open.
(ii) Yes, the intersection of finitely many open sets of intervals in ℝ is also open.
(iii) Yes, union of any collection of open sets of intervals in ℝ is also open.
Intuitively, it seems true to me. I just don't know how to prove it. I have practically no experience writing proofs.
I appreciate any help.