Lindelöf room

A Lindelöf space is a mathematical object from the set theoretical topology . It is a concept that generalizes that of compact space . The Lindelöf room is named after the mathematician Ernst Leonard Lindelöf .

definition

A topological space is called a Lindelöf space if every open cover has at most a countable partial cover .

Lindelof's Theorem

If the topological space has a countable basis , then it is a Lindelöf space. ${\displaystyle X}$ ${\displaystyle X}$

Other properties

• Every compact room is a Lindelöf room. More generally, every compact room is a Lindelöf room. ${\displaystyle \sigma }$
• A topological space is compact if and only if it is countably compact and Lindelöf space.
• For rooms that can be metrised , the three properties are equivalent to second countable , lindelöf and separable .
• Closed sub -rooms of Lindelöf rooms are again Lindelöf rooms.
• Any regular room that is a Lindelöf room is a normal room .

literature

• Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 .