Page 76 - DMTH503_TOPOLOGY

Topology

Notes 7.1.3 Open and Closed Map

Definition: Open Map
A map f : (X, T)  (Y, U) is called an open or interior map if it maps open sets onto open sets i.e.
if

any G  T  f(G)  U.
Definition: Closed Map
A map f : (X, T)  (Y, U) is called a closed map if

any T-closed set f  f(F) is U-closed set.

Example 4: (i) Let T denote the usual topology on R. Let a be any non-zero real number,
Then each of the following map is open as well as closed.
(a) f : (R, T)  (R, T) s.t. f(x) = a + x
(b) f : (R, T)  (R, T) s.t. f(x) = ax

In this case if a = 0, then this map is closed but not open.
(ii) The identity map f : (X, T)  (X, T) is open and as well as closed.
(iii) A map from an indiscrete space into a topological space is open as well as closed.
(iv) A map from a topological space into a discrete space is open as well as closed.

Note Proof of (i) b,
Let a  0 and A = (b, c)  T arbitrary.
Then f(b) = ab, f(c) = ac.
 f(A) = (ab, ac)  J
i.e., image of an open set is an open set under the map f(x) = ax, a  0. Hence this map is
open.

Similarly  f [b,c] = [ab,bc], i.e. image of a closed set is closed.
 f is a closed map
Consider the case in which a = 0
Then f(x) = ax = 0,  x  R

 f(x) = 0  x  R.

Now  f [b,c]  {0} A Finite set A   closed set for a finite set is a T-closed set.
Now the image of a closed set is closed and hence f is a closed map.
Again f (5, 6) = {0}  an open set.

 image of an open set is not open.
Consequently, f is not open.

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