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Separation Axioms

This quiz will test your understanding of separation axioms in topology.

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

Which of the following is a separation axiom?

  1. T0
  2. T1
  3. T2
  4. T3
  5. T4
Question 2 Multiple Choice (Single Answer)

In a T0 space, which of the following is true?

  1. Every point is closed.
  2. Every point is open.
  3. Every point is both open and closed.
  4. None of the above.
Question 3 Multiple Choice (Single Answer)

In a T1 space, which of the following is true?

  1. Every point is closed.
  2. Every point is open.
  3. Every point is both open and closed.
  4. Every point is either open or closed.
Question 4 Multiple Choice (Single Answer)

In a T2 space, which of the following is true?

  1. Every point is closed.
  2. Every point is open.
  3. Every point is both open and closed.
  4. Every two distinct points can be separated by open sets.
Question 5 Multiple Choice (Single Answer)

In a T3 space, which of the following is true?

  1. Every point is closed.
  2. Every point is open.
  3. Every point is both open and closed.
  4. Every two distinct points can be separated by open sets, and every closed set is a Gδ set.
Question 6 Multiple Choice (Single Answer)

In a T4 space, which of the following is true?

  1. Every point is closed.
  2. Every point is open.
  3. Every point is both open and closed.
  4. Every two distinct points can be separated by open sets, and every closed set is a Gδ set, and every point is a Gδ set.
Question 7 Multiple Choice (Single Answer)

Which of the following is an example of a T0 space that is not a T1 space?

  1. The Sorgenfrey line
  2. The Cantor set
  3. The Sierpinski space
  4. The real line with the usual topology
Question 8 Multiple Choice (Single Answer)

Which of the following is an example of a T1 space that is not a T2 space?

  1. The Sorgenfrey line
  2. The Cantor set
  3. The Sierpinski space
  4. The real line with the usual topology
Question 9 Multiple Choice (Single Answer)

Which of the following is an example of a T2 space that is not a T3 space?

  1. The Sorgenfrey line
  2. The Cantor set
  3. The Sierpinski space
  4. The real line with the usual topology
Question 10 Multiple Choice (Single Answer)

Which of the following is an example of a T3 space that is not a T4 space?

  1. The Sorgenfrey line
  2. The Cantor set
  3. The Sierpinski space
  4. The real line with the usual topology
Question 11 Multiple Choice (Single Answer)

Which of the following is an example of a T4 space?

  1. The Sorgenfrey line
  2. The Cantor set
  3. The Sierpinski space
  4. The real line with the usual topology
Question 12 Multiple Choice (Single Answer)

Which of the following is an example of a space that is not a T0 space?

  1. The Sorgenfrey line
  2. The Cantor set
  3. The Sierpinski space
  4. The real line with the usual topology
Question 13 Multiple Choice (Single Answer)

Which of the following is an example of a space that is not a T1 space?

  1. The Sorgenfrey line
  2. The Cantor set
  3. The Sierpinski space
  4. The real line with the usual topology
Question 14 Multiple Choice (Single Answer)

Which of the following is an example of a space that is not a T2 space?

  1. The Sorgenfrey line
  2. The Cantor set
  3. The Sierpinski space
  4. The real line with the usual topology
Question 15 Multiple Choice (Single Answer)

Which of the following is an example of a space that is not a T3 space?

  1. The Sorgenfrey line
  2. The Cantor set
  3. The Sierpinski space
  4. The real line with the usual topology
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