Sachchidanand Prasad

04-01-2024

Problem: Consider $\mathbb{R} ^2 $ with the usual metric. Let \begin{align*} A & = \left\{ (x,y) \in \mathbb{R} ^2 : x^2 + y^2 \leq 1 \right\} \\ B & = \left\{ (x,y) \in \mathbb{R} ^2 : (x - 2)^2 + y^2 \leq 1 \right\} . \end{align*} Let $M = A \cup B$ and $N = \mathrm{interior} (A) \cup \mathrm{interior}(B) $. Then which of the following statements is TRUE?
  1. $M$ and $N$ are connected
  2. Neither $N$ nor $N$ is connected
  3. $M$ is connected and $N$ is not connected
  4. $M$ is not connected and $N$ is connected
topology connectedness GATE-2022
Solution: Let us denote $\mathrm{interior}(A)$ by $\mathrm{int}(A)$. We will draw the two sets $M$ and $N$.
The set M

Set $M$

The set N

Set $N$

From the figures above, it is clear that $M$ is path connected and hence connected, whereas $N$ is disconnected. More precisely, $X = \mathrm{int}(A)$ and $Y = \mathrm{int}(B)$ give a separation of the set $N$. Thus, the correct answer will be (C).