Countable tightness is not discretely reflexive in $$\sigma$$-compact spaces

Abstract Answering a question raised by V.V. Tkachuk in [10], we present several examples of $$\sigma$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> -compact spaces, some only consistent and some in ZFC, that are not countably tight but in which the closure of any discrete subset is countably tight. In fact, in some of our examples the closures of all discrete subsets are even first countable.