Topology Quiz

Which of the following collections is a basis for the standard topology on R?

• {(a, b)∣a, b ∈ R, a< b}

• {[a, b]∣a, b ∈ R, a < b}

• {(a, b)∣a, b ∈ Q, a < b}

• {(a, b]∣a, b ∈ R, a < b}

In the standard topology on R, which of the following is NOT an open set?

• (0, 1)

• [0,1)

• n=1∞( 0, 1n )

• (−∞,5)

Which of the following subsets of R is dense in the standard topology?

• Z (integers)

• [0, 1]

• Q (rationals)

• (0, 1)

What is the closure of (0, 1) in the standard topology on R?

• (0, 1)

• [0, 1)

• [0, 1]

• (0, 1]

If B is a basis for a topology τ on X, which of the following is true?

• Every element of X belongs to exactly one basis element.

• Every open set in τ is a union of basis elements.

• The intersection of any two basis elements is always a basis element.

• Basis elements are always open sets.

Which of the following subsets of R is compact in the standard topology?

• (0, 1)

• [0, 1)

• [0, 1]

• (0, ∞)

Which of the following subsets of R is connected in the standard topology?

• (0,1)∪(2,3)

• (0,1)

• Q

• [0,1]∖{1/2}

Which of the following statements is true in a T1​-space?

• All open sets are closed.

• Every singleton set is closed.

• The space is compact.

• The space is connected.

A function f: R→R is continuous if:

• f-1(U) is open for every open set U ⊆ R.

• f (U) is open for every open set U ⊆ R.

• f-1(U) is closed for every closed set U⊆R.

• f (U) is closed for every closed set U⊆R.

Which of the following spaces has a countable basis?

• R with the standard topology

• R2 with the standard topology

• (0, 1) with the subspace topology from R

• All of the above