The transistor is off when there's no bias voltage or when the bias voltage is less than 0.7 V. The switch is on when the base is saturated so that collector current can flow without restriction.

In which region transistor acts as a switch?

Transistor acts as a switch in the saturation region and cutoff region. The emitter-base junction and the collector-base junction is reverse biased in the cutoff region. Both the junctions are forward biased in the saturation region.

When transistor works as an open switch?

A transistor will work as an open switch in the cut off state i.e., when both the emitter and collector are reversed biased.