# Suppose {Iα}α∈A is a collection of open intervals that are mutually disjoint: If α1 ≠ α2, then Iα1 ∩ Iα2 = ∅. Prove or disprove: A must be at most countable.

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Suppose {Iα}α∈A is a collection of open intervals that are mutually disjoint: If α1 ≠ α2, then Iα1 ∩ Iα2 = ∅. Prove or disprove: A must be at most countable.

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Step 1

Consider the given information:

The collection {{Iα}, where α belongs to A} of open intervals that are mutually disjoint.

Now, let S be the union of all such disjoint open interval:

Step 2

Since, all intervals are mutually disjoint, so above mentioned collection is a disjoint collection.

Now, to show that they form a countable collection:

Let {x1, x2, x3, ….} denote the countable set of rational numbers. In each interval say, Ix there will be infinitely many x, but among these there will be exactly one with...

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