Let X=Π∞n=1{0,1}. Show that X is not compact in the box topology. Whether X is compact in the product topology ?
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An arbitrary product of compact spaces is compact in the product topology.
If a family F of subsets of X has finite intersection property then for each α∈J,∩A∈Fπα(A) is non-empty.
Let (X,Ω) be a topological space with topology Ω. Let A,Y be subsets of X such that A⊆Y. Then
(1) A is closed in the subspace topology on Y iff A=B∩Y for some closed subset B of X.
(2) The closure of A in the subspace topology on Y equals ¯A∩Y.
x∈¯A iff either x∈A or x is a cluster point of A.
If X is a Hausdorff space then every neighbourhood of a cluster point of A contains infinitely many points from A.
There is no function g:(0,1)→R which is continuous on rationals and discontinuous on irrationals.
Rn is homeomorphic to R iff n=1.
Let f:A→Πmi=1Xi be given by f(a)=(f1(a),⋯,fm(a)), where the functions fi:A→Xi(i=1,⋯,m) are given. Then f is continuous iff each fi is continuous.
Let f:A→ΠαXα be given by f(a)=(fα(a)), where the functions fα:A→Xα are given. Suppose ΠαXα has product topology. Show that f is continuous iff each fα is continuous.
Let {xn=(xn1,xn2,⋯,)}be a sequence in the product space Πα∈IXα with product topology. Then the sequence {xn} converges tox+(x1,x2,⋯,)∈ΠαXα iff for every positive integer m,{πm(xn)=xnm} converges to xm.
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