Bulletin of the Belgian Mathematical Society - Simon Stevin

Discrete reflexivity in function spaces

V.V. Tkachuk

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We study systematically when $C_p(X)$ has a topological property $\mathcal{P}$ if $C_p(X)$ is discretely $\mathcal{P}$, i.e., the set $\overline {D}$ has $\mathcal{P}$ for every discrete subspace $D\subset C_p(X)$. We prove that it is independent of ZFC whether discrete metrizability of $C_p(X)$ implies its metrizability for a compact space $X$. We show that it is consistent with ZFC that countable tightness and Lindelöf $\Sigma$-property are not discretely reflexive in spaces $C_p(X)$. It is also established that a space $X$ must be countable and discrete if $C_p(X)$ is discretely Čech-complete. If $C_p(X)$ is discretely $\sigma$-compact then $X$ has to be finite.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 22, Number 1 (2015), 1-14.

First available in Project Euclid: 20 March 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54C35: Function spaces [See also 46Exx, 58D15] 54C05: Continuous maps
Secondary: 54G20: Counterexamples

function spaces pointwise convergence topology tightness discrete subspaces discretely reflexive property spread character discretely $\sigma$-compact space discretely Čech-complete space


Tkachuk, V.V. Discrete reflexivity in function spaces. Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 1, 1--14. doi:10.36045/bbms/1426856853. https://projecteuclid.org/euclid.bbms/1426856853

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