Toshiji Terada
Abstract
A Tychonoff space which contains no realcompact space as a dense subspace is constructed. Let $\mathcal {D}\mathcal {R}$ be the class of all spaces which contain some realcompact spaces as dense subspaces. Then, as a consequence of the above result, it follows that $\mathcal {D}\mathcal {R}$ is not closed-hereditary.
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A Tychonoff space which contains no realcompact space as a dense subspace is constructed. Let $\mathcal {D}\mathcal {R}$ be the class of all spaces which contain some realcompact spaces as dense subspaces. Then, as a consequence of the above result, it follows that $\mathcal {D}\mathcal {R}$ is not closed-hereditary.
Key concepts: Tychonoff space, Linear subspace, Subspace topology, Space (punctuation), Mathematics, Class (philosophy), Combinatorics, Pure mathematics