Topology Revision

Let X be a topological space.

True or false: any subset of X is either open or closed

• True

• False

The preimage of a path-connected space is path-connected.

• True

• False

How many cut points of type 2 does the letter Y have?

• 0

• 1

• 2

• at least 3

How many cut points of type 2 does ℝ²\{0} have?

• 0

• 1

• 2

• at least 3

Which ONE of the following is an OPEN subset of [0,1]?

• (-1,1)

• [0,0.1)

• { x in ℝ: |x-1|<0.5}

True or false: the image of an open subset under a continuous map is open.

• True

• False

Let X be given topologies t1 and t2.

True or false: the identity from (X, t1) to (X,t2) is always continuous

• True

• False

Every closed subset of a compact space is compact.

• True

• False

Every compact subset of a topological space is closed.

• True

• False

True or false: The discrete topology is always Hausdorff.

• True

• False