Understand the problem

Let X\subseteq R^2 be the subset s.tX=\{(x,y): x=0, |y| \leq 1\} \cup \{(x,y): 0<x\leq 1, y=\sin \frac 1x\}

Consider the following statements:

  • X is compact.
  • X is connected.
  • X is path-connected.

How many of the statements is/are true:

  1. 0
  2. 1
  3. 2
  4. 3
Source of the problem
TIFR 2019 GS Part A, Problem 17
Topic
Topology
Difficulty Level
Easy
Suggested Book
Topology, Munkres

Start with hints

Do you really need a hint? Try it first!

Consider the first part of the union. Is it bounded? Can you think about option 1)?
Can you think about the second part of the union? Do you remember the example about comb space which is a famous example of connected space but not path connected?
See this function in the second part of the union is a function of the same kind(comb space)(I will suggest you graph the function as well) but the first part of the union covers the problematic point at zero. Hence this is connected and path connected.

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