On Šapirovskiı̆ʹs theorem

We present a new version of Šapirovskiı̆ʹs well-known criterion for the existence of a continuous mapping from a compactum onto a Tychonoff cube. From this, we prove that under MA for every compact Hausdorff space X of weight less than 2ℵ0 and every infinite cardinal τ<2ℵ0 the following conditions are equivalent: 1. exists a continuous surjection from X onto the Tychonoff cube Iτ; exists a continuous injection from the Cantor cube Dτ into X; exists a closed subset Y⫅X such that χ(y,Y)≥τ for any y∈Y.