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We will show in this note that if      M M       is an      R ∞   - (or  Q ∞   - ) {R^\infty }{\text { - (or }}{Q^\infty }{\text { - )}}       manifold having the homotopy type of a finite complex, then the homeomorphism group of      M M      , endowed with the compact-open topology, has the local strong      U V ∞  U{V^\infty }      -property with respect to the classes of pseudo CW complexes      C W ( C ) \mathcal {C}\mathcal {W}(\mathcal {C})       and      C W ( M ) \mathcal {C}\mathcal {W}(\mathcal {M})      .

A local strong 𝑈𝑉^{∞}-property of the homeomorphism groups of 𝑅^{∞}- (𝑄^{∞}-) manifolds