Quotient maps onto submaximal spaces

Whyburn [19] showed that for a T 1 -space Y, every quotient map onto Y is pseudo-open (= hereditarily quotient) if and only if Y is an accessibility space defined in [19]. Similar topics to Whyburnʼs result were studied in Siwiec [15], Michael, Olson and Siwiec [13], Lin and Zhu [10] and so on. In this paper, advancing the results in [10], we show that for a T 1 -space Y: (1) every quotient map onto Y is almost-open if and only if every quotient map onto Y is strictly countably bi-quotient if and only if Y is submaximal and extremally disconnected, and (2) every quotient map onto Y is bi-quotient if and only if Y is submaximal and has property NCA ( ω ) defined in this paper. Similar topics on countable-covering maps are also studied.