Topologies T 1, T 2 on X are induced by metrics d 1,d 2, with T 1 T 2. Which is true?

Convergence in $(X,d_1)$ implies convergence in $(X,d_2)$.

Every open $d_1$-ball is an open $d_2$-ball.

The identity from $(X,d_1)$ to $(X,d_2)$ is continuous.

GATE 2020 MA – Question 11

Topology · Topological Spaces and Constructions · 1 mark · Multiple choice

Topologies $\mathcal T_1,\mathcal T_2$ on X are induced by metrics $d_1,d_2$, with $\mathcal T_1\subseteq\mathcal T_2$. Which is true?

Convergence in $(X,d_2)$ implies convergence in $(X,d_1)$.
Convergence in $(X,d_1)$ implies convergence in $(X,d_2)$.
Every open $d_1$-ball is an open $d_2$-ball.
The identity from $(X,d_1)$ to $(X,d_2)$ is continuous.

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Correct answer: (A) Convergence in $(X,d_2)$ implies convergence in $(X,d_1)$.

Explanation

The topology $\mathcal T_2$ is finer, so every $\mathcal T_1$ neighbourhood is also a $\mathcal T_2$ neighbourhood. A sequence eventually inside every $\mathcal T_2$ neighbourhood is therefore eventually inside every $\mathcal T_1$ neighbourhood. Thus A. The continuous identity goes from the finer topology to the coarser one, and equality of open sets does not require equality of metric balls.

GATE 2020 MA – Question 11

Explanation

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