Absolute and non-absolute $\mathcal{F}$-Borel spaces

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.