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COUNTABLE SPACES, REALCOMPACTNESS, AND THE PSEUDOINTERSECTION NUMBER
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 29 October 2024, pp. 1-17
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- Article
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All spaces are assumed to be Tychonoff. Given a realcompact space X, we denote by
$\mathsf {Exp}(X)$ the smallest infinite cardinal 
$\kappa $ such that X is homeomorphic to a closed subspace of 
$\mathbb {R}^\kappa $. Our main result shows that, given a cardinal 
$\kappa $, the following conditions are equivalent: • There exists a countable crowded space X such that
$\mathsf {Exp}(X)=\kappa $.•
$\mathfrak {p}\leq \kappa \leq \mathfrak {c}$.
$\mathfrak {d}\leq \kappa \leq \mathfrak {c}$, every countable dense subspace of 
$2^\kappa $ provides such an example. This will follow from our analysis of the pseudocharacter of countable subsets of products of first-countable spaces. Finally, we show that a scattered space of weight 
$\kappa $ has pseudocharacter at most 
$\kappa $ in any compactification. This will allow us to calculate 
$\mathsf {Exp}(X)$ for an arbitrary (that is, not necessarily crowded) countable space X.
