Add T be the class consisting of R,Q and all infinite, open intervals An=q,xAn∈Q,tQ, the rationals. Show that T is not a topology on R−.

We can find the sequence of rational number (qn)n such that qn&gt;2,n∈N and limqn=2n→∞ Notice that (qn,∞)∈T, and that ∞ U(qn1,∞)=(2,∞)n=1 (here we nedeed that qn&gt;2). However, (2,∞)∉T since 2 is not a rational number. Thus, T contains sets whose union is not in T, so T is not a topology.

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Answered question

2021-02-06

Add T be the class consisting of R,Q and all infinite, open intervals $A n = q, x A n \in Q, t Q$ , the rationals. Show that T is not a topology on R−.

Answer & Explanation

lamanocornudaW

Skilled2021-02-07Added 85 answers

We can find the sequence of rational number (qn)n such that $q n > \sqrt{2}, n \in N$
and
$lim q n = \sqrt{2} n \to \infty$
Notice that $(q n, \infty) \in T$ , and that $\infty U (q n 1, \infty) = (\sqrt{2}, \infty) n = 1$ (here we nedeed that $q n > \sqrt{2}$ ).

However, $(\sqrt{2}, \infty) \notin T$ since $\sqrt{2}$ is not a rational number. Thus, T contains sets whose union is not in T, so T is not a topology.

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