Problem regarding the one-point compactification

Reference no: EM13990726

1. Let (X, d) be a locally compact separable metric space. Show that its one-point compactification is metrizable.

2. Let X be any noncompact metric space, considered as a subset of its Tychonoff-C? ech compactification K . Let y ∈ K \X . Show that K is not metrizable by showing that there is no sequence xn ∈ X with xn → y in K . Hint: If xn → y, by taking a subsequence, assume that the points xn are all different. Then, {x2n }n≥1 and {x2n-1}n≥1 form two disjoint closed sets in X . Apply Urysohn's Lemma (2.6.3) to get a continuous function f on X with f (x2n ) = 1 and f (x2n-1) = 0 for all n. So {xn } cannot converge in K to y.

Reference no: EM13990726

Questions Cloud

Nondecreasing sequence of bounded stopping times : Let α be a stopping time and τ (n) a nondecreasing sequence of bounded stopping times, both for a martingale {Xn, Bn }n≥1. Show that α is also a stopping time for the martingale {Xτ (n), Bτ (n)}n≥1, in the sense that {α ≤ τ (n)}∈ Bτ (n) for each n..

Which compound will be accumulated by the double mutant : In Neurospora, two genes participate in the synthesis of valine. The wild-type allele for one gene is designated val-1+ and a mutant allele designated val-1-. The second gene has a wild-type val-2+ allele and a mutant val-2- allele. These two gene..

Determine the maximum pressure behind the dam : Fresh water is impounded behind a dam to a height h of 18 ft, as a shown. determine the maximum pressure behind the dam.

Problem regarding the real-valued function : Show that for any metric space S, if A ⊂ S and x ∈ A\ A, then there is a bounded, continuous real-valued function on A which cannot be extended to a function continuous on A∪{x }. Hint: f (t ) := sin(1/t ) for t > 0 cannot be extended continuously..

Problem regarding the one-point compactification : Let (X, d) be a locally compact separable metric space. Show that its one-point compactification is metrizable.

Contains real numbers of arbitrarily large : If S exists, then S(x, y) = 0 ∈ R. Then by Problem 4 there must be neighborhoods U of x and V of y in K such that S(u, v) ∈ R and |S(u, v)| 1 for all u ∈ U and v ∈ V . Show, however, that each of U and V contains real numbers of arbitrarily l..

Problem regarding the compact closure : Hint: Given any x ∈ X , let U be a neighborhood of x with compact closure. Show that there is a continuous real function f on X ,0 at x , and 1 on X \U . Use this function to show that x is not in the closure of K \U .

Cauchy net converges to some element : The uniform space is called complete if every Cauchy net converges to some element of S (for the topology of U ). If a metric space (S, d) is complete, prove it is also complete as a uniform space.

Find the speed of the block just after leaving the spring : A 2.00-kg block is pushed against a 400 N/m spring, compressing it 22.0 cm. When the block is released, it moves along a frictionless horizontal surface and then up an incline. The angle of the incline is 30.00 and the coefficient of kinetic frict..

User Account

All Pages