topological condition for a set to be uncountable


Let X be a nonempty compact Hausdorff space with no isolated points. To each finite 0,1-sequence α associate a point xα and an open neighbourhood Uα as follows. First, since X is nonempty, let x0 be a point of X. Second, since x0 is not isolated, let x1 be another point of X. The fact that X is Hausdorff implies that x0 and x1 can be separated by open sets. So let U0 and U1 be disjoint open neighborhoods of x0 and x1 respectively.

Now suppose for inductionMathworldPlanetmath that xα and a neighbourhood Uα of xα have been constructed for all α of length less than n. A 0,1-sequence of length n has the form (α,0) or (α,1) for some α of length n-1. Define x(α,0)=xα. Since x(α,0) is not isolated, there is a point in Uα besides x(α,0); call that point x(α,1). Now apply the Hausdorff property to find disjoint open neighbourhoods U(α,0) and U(α,1) of x(α,0) and x(α,1) respectively. The neighbourhoods U(α,0) and U(α,1) can be chosen to be proper subsetsMathworldPlanetmathPlanetmath of Uα. Proceed by induction to find an xαUα for each finite 0,1-sequence α.

Figure 1: The induction step: separating points in Uα.

Now define a function f:2ωX as follows. If α is eventually zero, put f(α)=xα. Otherwise, consider the sequence (x(α0),x(α0,α1),x(α0,α1,α2),) of points in X. Since X is compact and Hausdorff, it is closed and limit point compact, so the sequence has a limit pointPlanetmathPlanetmath in X. Let f(α) be such a limit point. Observe that for each finite prefix (α0,,αn) of α, the point f(α) is in U(α0,,αn).

Figure 2: Defining f(α) as a limit point of the x(α0,,αn).

Suppose α and β are distinct sequences in 2ω. Let n be the first position where αnβn. Then f(α)U(α0,,αn) and f(β)U(β0,,βn), and by construction U(α0,,αn) and U(β0,,βn) are disjoint. Hence f(α)f(β), implying that f is an injective function. Since the set 2ω is uncountable and f is an injective function from 2ω into X, X is also uncountable. ∎


The set [0,1] is uncountable.


Being closed and boundedPlanetmathPlanetmathPlanetmath, [0,1] is compact by the Heine-Borel Theorem; because [0,1] is a subspaceMathworldPlanetmathPlanetmath of the Hausdorff space , it too is Hausdorff; finally, since [0,1] has no isolated points, the preceding theorem implies that it is uncountable. ∎

Title topological condition for a set to be uncountable
Canonical name TopologicalConditionForASetToBeUncountable
Date of creation 2013-03-22 16:15:15
Last modified on 2013-03-22 16:15:15
Owner mps (409)
Last modified by mps (409)
Numerical id 15
Author mps (409)
Entry type Theorem
Classification msc 54D10
Classification msc 54A25
Classification msc 54D30