8 Finite products

Question 1

Product topology

Answer 1

Let (X1, T1), (X2, T2),…, (Xn, Tn) be topological spaces. Then the product topology T on the set X1 x X2 x … x Xn is the topology having the family {O1 x O2 x. … On, O_i e T_i, i = 1…n} as a basis. The set X1 x X2 x … x Xn with the topology T is said to be the product of the spaces (X1, T1), (X2, T2)…, (Xn, Tn) and is denoted by (X1 x X2 …, T)

Question 2

Finer / coarser topology

Answer 2

Let T1 and T2 be topologies on a set X. Then T1 is said to be a finer topology than T2 (and T2 is said to be a coarser topology) if T1 )_ T2

Question 3

Open mapping

Answer 3

Let (X, T) and (Y, T1) be topological spaces and f a mapping from X into Y. Then f is said to be an open mapping if for every A e T, f(A) e T1.

Question 4

Closed mapping

Answer 4

Let (X, T) and (Y, T1) be topological spaces and f a mapping from X into Y. The mapping is said to be a closed mapping if for every closed set B in (X, T), f(B) is closed in (Y, T1).

Question 5

The component of a point

Answer 5

Let (X, T) be a topological space and let x be any point in X. The component in X of x, C_X(x), is defined to be the union of all connected subsets of X which contain x.

Question 6

Continuum

Answer 6

A topological space is said to be a continuum if it is compact and connected.