## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

### Discrete reflexivity in function spaces

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 20 March 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1426856853

**Mathematical Reviews number (MathSciNet)**

MR3325716

**Zentralblatt MATH identifier**

1316.54007

**Subjects**

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

Secondary: 54G20: Counterexamples

**Keywords**

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

#### Citation

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