Let M M be a compact connected Hilbert cube manifold ( Q Q -manifold). Define C z ( M ) {C_z}\left ( M \right ) to be the smallest integer k k such that M M can be covered with k k open subsets each of which is homeomorphic to Q × [ 0 , 1 ) Q \times \left [ {0,1} \right ) . Recently L. Montejano proved that, for every compact connected polyhedron P , C z ( P × Q ) = cat ⁡ ( P ) + 1 P,{C_z}\left ( {P \times Q} \right ) = \operatorname {cat}\left ( P \right ) + 1 , where cat ⁡ ( P ) \operatorname {cat} \left ( P \right ) is the Lusternik-Schnirelmann category of P P . Using a different approach, we prove a noncompact analog of the above theorem by showing that C z ( P × Q × [ 0 , 1 ) ) = cat ⁡ ( P ) {C_z}\left ( {P \times Q \times \left [ {0,1} \right )} \right ) = \operatorname {cat}\left ( P \right ) for every compact connected polyhedron P P .