In this paper we study operators originated from semi-B-Fredholm theory and as a consequence we get some results regarding boundaries and connected hulls of the corresponding spectra. In particular, we prove that a bounded linear operator $T$ acting on a Banach space, having topological uniform descent, is a { \bf BR} operator if and only if $0$ is not an accumulation point of the associated spectrum $\sigma_{ \bf R}(T)=\{ \lambda\in\CC:T-\lambda I\notin { \bf R}\}$, where ${ \bf R}$ denote any of the following classes: upper semi-Weyl operators, Weyl operators, upper semi-Fredholm operators, Fredholm operators, operators with finite (essential) descent and ${ \bf BR}$ the B-regularity associated to ${ \bf R}$ as in \cite{ P8}. Under the stronger hypothesis of quasi-Fredholmness of $T,$ we obtain a similar characterization for $T$ being a { $\bf BR$} operator for much larger families of sets ${ \bf R}.$
Categories: math.SP