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TIFR-2019-Maths-A: 17

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Let $X\subset \mathbb{R}^{2}$ be the subset

$$X=\left \{ \left ( x,y \right ) \left | x=0, \right |y \mid \leq 1\right \}\cup \left \{ \left ( x,y \right ) \mid 0 < x \leq 1, y=sin \frac{1}{x}\right \}.$$

Consider the following statements:

  1. $X$ is compact
  2. $X$ is connected
  3. $X$ is path connected

How many of the statements (i)-(iii) is /are true?

  1. $0$
  2. $1$
  3. $2$
  4. $3$
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