Exercise 6, § 4

Let X be a topological space.

(a) Let x be a point of X and U an open neighbourhood of x with only a finite number of irreducible components. Show that there exists a neighbourhood V of x such that every neighbourhood contained in V is connected.

(b) Suppose that every point of X possesses an open neighbourhood with only a finite number of irreducible components. Show that the following Properties are equivalent:

(α) The irreducible components of X are open. (β) The irreducible components of X are identical with its connected components.

(γ) The connected components of X are irreducible. (δ) Two distinct irreducible components of X do not meet.