If X is second countable then every open covering of X contains a countable subcover.
Adv.
Let (X,d) be a metric space with metric. Let A be a subset of X, and for x∈X, let d(x,A):=inf{d(x,a):a∈A}. Show that d(x,A) is a continuous function of x.
Let (X,Ω) be a topological space with the property that any two disjoint non empty closed subsets of X can be separated by a continuous function. Show that for any disjoint non-empty closed subsets A and B of X, there exist disjoint open sets U and V such that A⊆U and B⊆V.
Consider the topological space RK . Show that there are no disjoint open sets U and V of RK such that {0}⊆U and K⊆V. In particular, RK is Hausdorff but not normal.
Whether or not any two disjoint non-empty closed subsets of a normal space X can be separated by a continuous function ?
Suppose X admits a family {Ur}r∈Q of nested neighbourhoods. Definef:X→[0,1] by f(x)=infQ(x). Verify the following:
(1) f(a)=0 for every a∈A.
(2)f(b)=1 for every b∈B.
(3) f(x)≤rr for any x∈¯Ur,
(4) f(x)≥r for any x∉Ur.
Let R be a commutative ring. Prove that HOMR(R,R)) and R are isomorphic as rings.
Given closed non-empty disjoint subsets A and B of a normal space X, there exists a continuous function f:X→[0,1] such that f|A=0 and f|B=1.
Let X be a compact Hausdorff space without any isolated points. Consider the vector space C(X) of continuous functions f:X→R.
Show that C(X) is infinite-dimensional.
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 ?
An arbitrary product of compact spaces is compact in the product topology.
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