We prove that if there is a model of set-theory which contains no first countable, locally compact, scattered, countably paracompact space X, whose Tychonoff square is a Dowker space, then there is an inner model which contains a measurable cardinal.

For a Tychonoff space $X$, we denote by $C_p(X)$ the of all real-valued continuous functions on $X$ with topology pointwise convergence.  We study functional characterization covering property Hurewicz.

We define branch product topologies, a new product construction. Branch product topologies generalize sum topologies and product topologies. We investigate criteria for branch products or iterations of branch products to preserve various topological properties, focusing on the separation axioms and compactness. In particular, we generalize the Tychonoff Theorems for compactness and D-compactness.

Article history: Received 22 January 2009 Accepted 1 December 2009 MSC: primary 05A18, 03E05, 54A10 secondary 03E35, 54A25, 05D05

If B is a compact space and B \ {pt} is Lindelöf then Bκ \ { − → pt} is star-Linedlöf for every κ. If B \ {pt} is compact then Bκ \ { − → pt} is discretely star-Lindelöf. In particular, this gives new examples of Tychonoff discretely star-Lindelöf spaces with unlimited extent.