Let X and Y be topological spaces and Cx y the set of all co

Let X and Y be topological spaces, and C(x, y) the set of all continuous maps from X to Y. We can place a topology on C(X, Y), called the weak topology, as follows: for every x belongsto X there is a function ev_x: C(X, Y) rightarrow Y called evaluation at x which takes a continuous function f: X rightarrow Y and evaluates f at x: ev_x(f)=f(x). The weak topology on C(X, Y) is the topology generated by {ev_x)_x belongsto X. If Y is a topological space, compute the topological space C({*}, Y), i.e., compute c){*}, Y) as a set, and compute its topology.

Solution

It is assumed that {*} is a pointed space. (as this has not been defined in the problem)

C({*},Y) is just the set of all functions from a pointed space to Y.

As a set this may be identified with Y itself.

The weak topology on C({*},Y) is such that the map Id:Y->Y (the identity map) is continuous.

So the topology on Y as a set is identical to the topology on Y .