  Let $(X,J) $ be an almost complex manifold with a (smooth) involution
$\sigma:X\to X$ such that fix($\sigma$) is non-empty. Assume that $\sigma$ is a
complex conjugation, i.e, the differential of $\sigma$ anti-commutes with $J$.
The space $P(m,X):=\mathbb{S}^m\times X/\!\sim$ where $(v,x)\sim
(-v,\sigma(x))$ was referred to as a generalized Dold manifold. The above
definition admits an obvious generalization to a much wider class of spaces
where $X, S$ are arbitrary topological spaces. The resulting space $P(S,X)$
will be called a generalized Dold space. When $S$ and $X$ are CW complexes
satisfying certain natural requirements, we obtain a CW-structure on $P(S,X)$.
Under certain further hypotheses, we determine the mod $2$ cohomology groups of
$P(S,X)$. We determine the $\mathbb Z_2$-cohomology algebra when $X$ is (i) a
torus manifold whose torus orbit space is a homology polytope, (ii) a complex
flag manifold. One of the main tools is the Stiefel-Whitney class formula for
vector bundles over $P(S,X)$ associated to $\sigma$-conjugate complex bundles
over $X$ when the $S$ is a paracompact Hausdorff topological space, extending
the validity of the formula, obtained earlier by Nath and Sankaran, in the case
of generalized Dold manifolds.
