Connectedness

Which of the following is a connected space?

• The set of rational numbers, Q, with the standard topology.

• The interval [0,1] in R with the standard topology.

• The union of two disjoint open intervals in R.

• The set of integers, Z, with the standard topology.

A space X is connected if and only if:

• It can be written as the union of two non-empty disjoint open sets.

• It cannot be written as the union of two non-empty disjoint open sets.

• It contains at least one limit point.

• It is compact.

The component of a point x in a topological space X is:

• The largest connected subset of X containing x.

• The smallest connected subset of X containing x.

• The set of all points in X that are path-connected to x.

• The set of all points in X that are limit points of x.

In a discrete topological space, the components are:

• The entire space

• The individual points

• The open sets

• The closed sets

A space X is locally connected if:

• Every point has a connected neighborhood.

• Every point has a path-connected neighborhood

• Every point has a compact neighborhood

• Every point has a neighborhood that is both open and closed

If X is a connected space and f:X→Y is a continuous function, then:

• f(X) is always connected.

• f(X) is always compact.

• f(X) is always open.

• f(X) is always closed.

In a topological space X, the number of components is:

• Always finite.

• Always countable.

• Always uncountable.

• be finite, countable, or uncountable.

Which of the following is true

• Arbitrary cartesian product of connected space is connected

• Finite cartesian product of connected space is connected

• Countable cartesian product of connected space is connected

Let a space X is locally connected and if for every open set U of X then each component of U is . . . . .

• open in X

• closed in X

• neither open nor closed

Say True or False: If X is locally path connected, then the components and path components of X are the same.

• True

• False