For any 𝑋, the product 𝑋×𝑌 has remote points for some 𝑌

Peters, Thomas J.

doi:10.1090/S0002-9939-1985-0810178-5

For any $X$, the product $X\times Y$ has remote points for some $Y$
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Proc. Amer. Math. Soc. 95 (1985), 641-648 Request permission

Abstract:

Any space with a $\sigma$-locally finite $\pi$-base will be called a $\sigma - \pi$ space. The work of Chae and Smith can be extended to show that every nonpseudocompact $\sigma - \pi$ space has remote points.$^{2}$ Sufficient conditions for a product to be a $\sigma - \pi$ space are developed. It is shown that, for each space, if $\alpha$ is a cardinal with the discrete topology, where $\alpha$ is not less than $\pi$-weight of $X$, then $X \times {\alpha ^\omega }$ has remote points. Cardinal function criteria are developed for the existence of $\sigma - \pi$ spaces. An example is given of a pathological product which is a $\sigma - \pi$ space even though none of its finite partial products is a $\sigma - \pi$ space.

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