  We investigate $\mathcal F$-Borel topological spaces. We focus on finding out
how a~complexity of a~space depends on where the~space is embedded. Of
a~particular interest is the~problem of determining whether a~complexity of
given space $X$ is absolute (that is, the~same in every compactification of
$X$). We show that the~complexity of metrizable spaces is absolute and provide
a~sufficient condition for a~topological space to be absolutely $\mathcal
F_{\sigma\delta}$. We then investigate the~relation between local and global
complexity. To improve our understanding of $\mathcal F$-Borel spaces, we
introduce different ways of representing an~$\mathcal F$-Borel set. We use
these tools to construct a~hierarchy of $\mathcal F$-Borel spaces with
non-absolute complexity, and to prove several other results.
