Exercise · ZW-TOPOLOGICAL-SPACE-007 · ★★☆ Intermediate
In \(\mathbb{N}\), let \(\mathcal{T}\) be the family made of \(\varnothing\) and the sets whose complement is finite. Which one is open?
Choose one option.
A \(\mathbb{N}\setminus\{1,2\}\) B \(\{1,2\}\) C the set of even numbers D the set of prime numbers Show answer A. \(\mathbb{N}\setminus\{1,2\}\)← Back to the question
Explanation A nonempty open set must have a finite complement. The complement of \(\mathbb{N}\setminus\{1,2\}\) is \(\{1,2\}\), which is finite; in the others the complement is infinite.
A topological space is a set \(X\) together with a topology \(\mathcal{T}\) on it; the members of \(\mathcal{T}\) are its open sets \(U\).
\[(X,\mathcal{T}),\qquad U\text{ open}\iff U\in\mathcal{T}\]
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