# “Let Ƭ consist of ∅, ℝ, and all intervals (−∞, p) for p ∈ ℝ. Prove that Ƭ is a topology on ℝ.”

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.

• Why does it seem true to you? Seriously, trying to honestly answer this question for yourself is what this exercise is all about. – uniquesolution Jun 6 at 18:45

(i) It makes no sense to assert that a set is (or isn't) open before you have a topology. What you should say here is that $$\emptyset,\mathbb R\in T$$.

(ii) Let $$F$$ be a finite set of elements of $$T$$. You want to prove that its intersection belongs to $$T$$ too. Then:

• If $$\emptyset\in F$$, then the intersection is $$\emptyset$$, which belongs to $$T$$.
• If all elements of $$F$$ are equal to $$\mathbb R$$, then the intersection is $$\mathbb R$$, which belongs to $$T$$.
• If some, but not all, elements of $$F$$ are equal to $$\mathbb R$$, then the intersection is equal to the intersection of those elements of $$F$$ which are not equal to $$\mathbb R$$.
• If neither $$\emptyset$$ nor $$\mathbb R$$ belong to $$F$$, then each element of $$F$$ is of the form $$(-\infty,x_k)$$, with $$k\in\{1,2,\ldots,\#F\}$$ and $$x_k\in\mathbb R$$. Then the intersection is $$(-\infty,\min_kx_k)\in T$$.

(iii) Can you deal with it now?

• Under (ii), - First bullet: Is it ∅∩T, or is it ∅∩F, or does it not matter? - Second bullet: is it R∩T, or is it R∩F? - Fourth bullet: sorry, can you break this down? Why does each element of T take that form if neither ∅ nor R belong to T? Can you also explain what minkxk means? Appreciate your help – Math_Student_1 Jun 6 at 19:59
• Please read my answer again. I've edited it, since I wrote $T$ instead of $F$ four times. Sorry about that. – José Carlos Santos Jun 6 at 20:11