  We investigate the Wightman function, the vacuum expectation values of the
field square and the energy-momentum tensor for a scalar field with general
curvature coupling parameter in $(D+1)$-dimensional de Sitter spacetime with an
arbitrary number of compactified spatial dimensions. Both cases of periodicity
and antiperiodicity conditions along the compactified dimensions are
considered. Recurrence formulae are derived which express the vacuum
expectation values for the dS spacetime of topology $\mathrm{R}^{p}\times
(\mathrm{S}^{1})^{q}$ in the form of the sum of the vacuum expectation values
in the topology $\mathrm{R}^{p+1}\times (\mathrm{S}^{1})^{q-1}$ and the part
induced by the compactness of the $(p+1)$th spatial dimension. The behavior of
the topological parts is investigated in various asymptotic regions of the
parameters. In the early stages of the cosmological evolution the topological
parts dominate the contribution in the expectation values due to the
uncompactified dS part. In this limit the behavior of the topological parts
does not depend on the curvature coupling parameter and coincides with that for
a conformally coupled massless field. At late stages of the cosmological
expansion the expectation values are dominated by the part corresponding to
uncompactified dS spacetime. The vanishing of the topological parts is
monotonic or oscillatory in dependence of the mass and the curvature coupling
parameter of the field.
