On two problems in extension theory

In this note we introduce the concept of a quasi-finite complex. Next, we show that for a given countable simplicial complex L the following conditions are equivalent:
• L is quasi-finite.
• There exists a [L]-invertible mapping of a metrizable compactum X with e-dimX⩽[L] onto the Hilbert cube.Finally, we construct an example of a quasi-finite complex L such that its extension type [L] does not contain a finitely dominated complex.