Multicoherence of spaces of the form $X/M$

Abstract:

Let $X$ be a connected, locally connected, normal ${T_1}$-space and let $M$ be a closed connected, locally connected subspace of $X$. Suppose that $X/M$ denotes the space obtained by identifying $M$ in a single point, and that, for a connected space $Y$, $\imath (Y)$ denotes the multicoherence degree of $Y$. In this paper, we prove that if $M$ is unicoherent, then $\imath (X) = \imath (X/M)$. As an application of this result we prove that if $X = A \cup B$, where $A,B$ are closed subsets of $X$ and $A \cap B$ is connected, locally connected and unicoherent, then $\imath (X) = \imath (A) + \imath (B)$. Also, we prove that if $X/M$ is unicoherent, then $\imath (X) \leqslant \imath (M)$.