The theorem is remarkable, and gives some apparently counter-intuitive examples of spaces homeomorphic to the usual Q. Consider the “Sorgenfrey topology on Q,” which has the collection {(p, q] : p, q ∈ Q} as a base for its topology. This topology on Q is strictly finer than, and yet homeomorphic to, the usual topology of Q. Another example is Q×Q as a subspace of the Euclidean plane. In this article, we present three proofs of Sierpinski’s theorem.

A 1910 theorem of Brouwer characterizes the Cantor set as the unique totally disconnected, compact metric space without isolated points. A 1920 theorem of Sierpinski characterizes the rationals as… Expand

ABSTRACT This note provides a short, self-contained proof of the famous fact that any countable metric space without isolated points is homeomorphic to the space of rational numbers. The discussion… Expand

Abstract We study properties of the Golomb topology on polynomial rings over fields, in particular trying to determine conditions under which two such spaces are not homeomorphic. We show that if K… Expand

In der vorliegenden Masterarbeit wird das Problem des separablen Quotienten fur lokal-konvexe Raume der Form Cp(X), welches immer noch ungelost ist, behandelt. Der Zusammenhang mit einem weiteren… Expand

This master thesis deals with the separable quotient problem for locally convex spaces of the form Cp(X) which is still open. Its connection with another open problem from topology, namely Efimov’s… Expand