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In this note we investigate the question: when does the metrizability of a dense subspace of a topological space imply the metrizability of the whole space? We show that certain conditions always fail to be sufficient and then we examine some elementary examples. We conclude with a theorem which states that a first countable, regular, Hausdorff space $Y$ which has an open metrizable (in the subspace topology) subspace $X$ is metrizable provided $Y-X$ is scattered in $Y$. Our investigation is conducted on an elementary level.
This article focuses on the concept of primality, a topic which extends from the dawn of history to the present. It likewise foreshadows some of the challenges confronting the mathematical world of the twenty-first century. Various tests of primality are often cumbersome or difficult to apply - including the Sieve of Eratosthenes and Wilson's Theorem. Other tests are typified by Fermat's Little Theorem. The notion of repunit numbers extends this pursuit and leads to the intriguing area of fraudulent primes. It likewise provides an interesting classroom activity in which converses and the expressing of necessary and sufficient conditions are analyzed.