A property similar to paracompacness Definitaion1: A family $\{A_t\}_{t\in S}$ of subsets of a
topological space $X$ is locally finite if for every point $x\in
X$ there exists a neighborhood $U$ of $x$ such that the set
$\{s\in S : U \cap A\neq \emptyset\}$ is finite.
Dfinition2: A topological space $X$ is called a *-space
space if $X$ is a Hausdorff space and every open cover of $X$ has
a locally finite subcover.
Are *-spaces famous? Or are there any equivalence condition for
them? (Note that they are not paracompact)

Even more can be said, suppose $X$ has the property that every open cover $\mathcal{U}$ has a point-finite subcover. Then $X$ is compact. It's clear that $\ast$-spaces have this property (as locally finite implies point-finite).
Proof: let $\mathcal{U}$ be any open cover of $X$. Let $U_0$ be any non-empty open set from $\mathcal{U}$ (and $p \in U_0$). Define the open cover 
$$\mathcal{V} = \{ U \cup U_0: U \in \mathcal{U} \}\text{.}$$
Clearly $\mathcal{V}$ is also an open cover of $X$ so by assumption has a point-finite subcover $\mathcal{V}'$. As every member of $\mathcal{V}$ contains $p$, this subcover can only be point-finite if it is finite. But then finitely members of $\mathcal{U}$ also cover $X$, showing $X$ is compact. 
So demanding subcovers instead of refinements reduces almost all such variations to plain old compactness.
