A topological space is called compact if every open cover of contains a finite partial cover .
An overlap is called punctual if every point of the space lies in at most finitely many overlap sets. A topological space is called metacompact if every open cover has a point refinement.
An overlap is called locally finite if every point of the space has a neighborhood that intersects at most a finite number of overlap sets. As is well known, a topological space is called paracompact if every open cover has a locally finite refinement.
A cover is called -local finite , if it can be written as a countable union of families of sets , so that each point of space has a neighborhood for each , which cuts out at most finitely many sets .
A cover is called -discreet if it can be written as a countable union of families of sets , so that for each point and for each there is a neighborhood of this point which cuts out at most one of the sets . The -discrete and -local coverages play an important role in the Bing-Nagata-Smirnov theorem .
A T 1 -space is normal if and only if every open locally finite cover has a shrinkage.