If Y is compact show that the projection X x Y X is a close

If Y is compact show that the projection: X x Y --> X is a closed map. ( i.e. the projection maps closed sets (in X x Y) to closed sets in X for all closed sets in (X x Y) )

Solution

Let DD be a directed set and (xd)dD(xd)dD be a net in XX.

We can topologize Y=D{}Y=D{}, where DD, in a natural way: All points of DD will be isolated. Basic neighborhoods of are the sets of the form {}{xd;dd0}{}{xd;dd0} for d0Dd0D. (The reason that this seems to be relatively natural choice is that xdxd converges to xx in XX if and only if (xd,d)(xd,d) converges to (x,)(x,) in X×YX×Y.)

We want to show that the net (xd)dD(xd)dD has a cluster point. Let us denote A={(xd,x);dD}A={(xd,x);dD}. Since the map pYpY is closed, we have pY[A¯]pY[A¯]. This means that there is an xXxX such that (x,)A¯¯¯¯(x,)A¯.

Notice that basic neighborhoods of the point (x,)(x,) are of the form

U×{dD;dd0}U×{dD;dd0}

where d0Dd0D and UU is an neighborhood of xx.

Since every set of this form has nonempty intersection with AA we get that for each neighborhood UUof xx and for each d0d0 there exists dd0dd0 such that xdUxdU. Hence xx is a cluster point of the net (xd)dD(xd)dD.