Let T 1, T 2, T 3 be the smallest topologies on R containing, respectively, S 1=\(a,a+/n):a Q,n N\, S 2=\(a,b):a,b Q,a<b\ and S 3=\(a,b):a,b R,a<b\. Which holds?

$\mathcal T_3\supsetneq\mathcal T_1$

$\mathcal T_3\supsetneq\mathcal T_2$

$\mathcal T_1\supsetneq\mathcal T_2$

GATE 2020 MA – Question 48

Topology · Topological Spaces and Constructions · 2 marks · Multiple choice

Let $\mathcal T_1,\mathcal T_2,\mathcal T_3$ be the smallest topologies on R containing, respectively, $S_1=\{(a,a+\pi/n):a\in\mathbb Q,n\in\mathbb N\}$, $S_2=\{(a,b):a,b\in\mathbb Q,a<b\}$ and $S_3=\{(a,b):a,b\in\mathbb R,a<b\}$. Which holds?

$\mathcal T_3\supsetneq\mathcal T_1$
$\mathcal T_3\supsetneq\mathcal T_2$
$\mathcal T_1=\mathcal T_2$
$\mathcal T_1\supsetneq\mathcal T_2$

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Correct answer: (C) $\mathcal T_1=\mathcal T_2$

Explanation

Every displayed interval is ordinarily open, so all generated topologies are contained in the Euclidean topology. Rational-endpoint intervals form a Euclidean basis. The S₁ intervals also form a basis: within any neighbourhood of x choose n with π/n sufficiently small and a rational left endpoint a with a < x < a + π/n contained in that neighbourhood. Thus all three topologies are equal and C is correct.

GATE 2020 MA – Question 48

Explanation

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