For any space X, either $$C_p(X)$$ or $$C_pC_p(X)$$ is $$\psi $$-separable

Abstract We establish that, for any Tychonoff space X , at least one of the spaces $$C_p(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and $$C_pC_p(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> has a dense subspace of countable pseudocharacter. Under MA, we give an example of a space X such that $$C_p(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> does not have a dense subspace of countable functional tightness. We also show that there exists a compact zero-dimensional space K such that $$C_p(K,\{0,1\})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is exponentially separable while K is not a Sokolov space. For compact scattered spaces K of countable dispersion index, we show that $$C_p(K)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> has a dense exponentially separable subspace if and only if K is $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -monolithic. Our results provide answers to several published open questions.