Does Top category has both (epi, extremal mono) and (extremal epi, mono) factorization property? Let ${\bf{Top}}$ be a category of topological spaces with continuous functions. Does ${\bf{Top}}$ share extremal epi-mono factorization (and epi-extremal mono factorization) property? If yes, are these factorization the same? Or, at least, are they somehow related?

An extremal mono in $\mathbf{Top}$ is just an embedding and an extremal epi is just a quotient map.  So both factorizations you as for exist: the epi-extremal mono factorization of $f:X\to Y$ is the factorization $X\to I\to Y$ where $I\to Y$ is the inclusion of the image of $f$ as a subspace of $Y$, and the extremal epi-mono factorization is the factorization $X\to Q\to Y$ where $X\to Q$ is the quotient map for the quotient topology induced by $f$ on its image.  These two factorizations are the same iff $f$ is a quotient map onto its image.  In general, there is a canonical map $Q\to I$ which is compatible with the factorizations and this map is a continuous bijection, but not necessarily a homeomorphism.
