# Is this set open? From topology without tears book

This question is from "topology without tears" book, page 30. Example 1.2.3

$\{1\}$ has as complement $\mathbb{N}\setminus \{1\}$ , so it is not finite . ($\mathbb{N}$ is set of natural numbers with the cofinite topology) Then the book concludes that $\{1\}$ is not open!?

My question is : where this conclusion comes from? Does it mean that if complement of a set is infinite then that set is not open?

• well, what topology has $\mathbb{N}$ equipped on it? – Riccardo Oct 31 '15 at 21:35
• For example, if $\mathbb{N}$ has the COFINITE topology on it (which seems the case) then by definition of cofinite topology, a subset whose complementary is NOT finite, then it is NOT open. But still, it depends on your hypothesis – Riccardo Oct 31 '15 at 21:37