limit points of uncountable subsets of R^n

PropositionPlanetmathPlanetmath. Let n be an n-dimensional, real normed space and let An. If A is uncountable, then there exists limit pointPlanetmathPlanetmath of A in n.

Proof. For any k let


i.e. 𝔹k is a closed ballPlanetmathPlanetmath centered in 0 with radius k. Assume, that for any k the set


is finite. Then Vk=A would be at most countableMathworldPlanetmath. ContradictionMathworldPlanetmathPlanetmath, since A is uncountable. Thus, there exists k0 such that Vk0 is infiniteMathworldPlanetmath. But Vk0𝔹k0 and since 𝔹k0 is compactPlanetmathPlanetmath (and Vk0 is infinite), then there exists limit point of Vk0 in n. This completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Corollary. If An is uncountable, then there exist infinitely many limit points of A in n.

Proof. Assume, that there are finitely many limit points of A, namely x1,,xkn. For ε>0 define


Briefly speaking, Aε is a complementPlanetmathPlanetmath of a union of closed balls centered at xi with radii ε. Of course Aε since there are finitely many limit points. Let


Assume, that Vε is countable for every ε. Then


would be at most countable (of course under assumptionPlanetmathPlanetmath of Axiom of ChoiceMathworldPlanetmath). Contradiction. Thus, there is γ>0 such that Vγ is uncountable. Then (due to proposition) there is a limit point xn of Vγ. Note, that


for some 0<γ<γ. Thus x is different from any xi. Contradiction, since x is also a limit point of A.

Title limit points of uncountable subsets of R^n
Canonical name LimitPointsOfUncountableSubsetsOfRn
Date of creation 2013-03-22 19:07:57
Last modified on 2013-03-22 19:07:57
Owner joking (16130)
Last modified by joking (16130)
Numerical id 6
Author joking (16130)
Entry type Theorem
Classification msc 54A99