Monotone countable paracompactness

We study a monotone version of countable paracompactness, MCP, and of countable metacompactness, MCM. These properties are common generalizations of countable compactness and stratifiability and are shown to relate closely to the generalized metric g-functions of Hodel: MCM spaces coincide with β-spaces and, for q-spaces (hence first countable spaces) MCP spaces coincide with wN-spaces. A number of obvious questions are answered, for example: there are “monotone Dowker spaces” (monotonically normal spaces that are not MCP); MCP, Moore spaces are metrizable; first countable (or locally compact or separable) MCP spaces are collectionwise Hausdorff (in fact we show that wN-spaces are collectionwise Hausdorff). The extent of an MCP space is shown to be no larger than the density and the stability of MCP and MCM under various topological operations is studied.