This property has been called hypocompactness; it is strictly weaker than compactness and strictly stronger than paracompactness. The Sorgenfrey line is hypocompact but not compact: every open cover has a disjoint clopen refinement. Any hedgehog space $X$ of uncountable spininess is an example of a paracompact space that is not hypocompact. $X$ is metrizable, so it’s paracompact. If $X$ has spininess $\kappa$, $p$ is the centre point of the hedgehog, the points at the ends of the spines are $q_\xi$ for $\xi&lt;\kappa$, and the metric $d$ is as in the linked article, the open cover

$$\left\{B_d\left(p,\frac23\right)\right\}\cup\left\{B_d\left(q_\xi,\frac23\right):\xi&lt;\kappa\right\}$$

has no open refinement that is star-finite at $p$.

Added: Henno reminds me that I forgot to answer the last question. Paracompact Lindelöf spaces are hypocompact, so separable metric spaces are hypocompact.

Is this strengthening of paracompactness known?
I've created a little game on Android. Now, I want to add a multiplayer mode which works with bluetooth or WiFi, so you can play together with other "smartphones" in your neighborhood . Which is the best way to offer a fast, good working mulitplayer mode that supports so many devices as possible? I heard about Alljoyn, but Im not quite sure if it fits my needs. Which way do you recommend?

regards Lisa
