Verify that Bb and Bp are basis for a topology on ΠmαXα:
(1) Bb={Πα∈IUα:UαisopeninXα} .
(2) Bb={Πα∈IUα∈Bb:Uα=Xα for finitely many values of α}.
Adv.
Suppose that each Xα contains a non-empty proper open subset Uα. Show that ΩBp=ΩBb iff I has finite cardinality.
Show that the product topology is the smallest topology which makes all projections πα continuous.
Let f:A→ΠαXα be given by f(a)=(fα(a)), where the functions fα:A→Xα are given. Suppose ΠαXα has either box topology or product topology. Show that if f is continuous then each fα is continuous.
Show that D:Rω×Rω→R given by
D((xn),(yn)):=supnmin{|xn−yn|,1}n
defines a metric on Rω . Verify further that D≤du , where du is the uniform metric.
Show that any finite set is compact.
Let K⊆Y⊆X. Show that K is compact in X iff K is compact in Y.
We invoke a classical result from Analysis, which characterizes all compact subsets of Rn .
Let p be a polynomial in the real variables x,y. Then the zero set Z(p) of p is always closed. However, Z(p) may or may not be compact. For instance, if p(x,y)=x2=y2−1then Z(p) being the unit circle is compact. If p(x,y)=x then Z(p) is the Y -axis, which is certainly non-compact.
Let p be a non-constant polynomial in the complex variables z1,⋯,zm. Show that the zero set Z(p) of p is compact iff m=1.
Show that every subset of R with co-finite topology is compact.
Show that the continuous image of a compact space is again compact.
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