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

elementary-set-theorygeneral-topologyproof-writing

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.

Best Answer

(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?

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