Math Is Fun Forum

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Ernst Lindelöf

Ernst Lindelöf was a Finnish mathematician. Lindelöf spaces in general topology are named after him. A Lindelöf space is a toplogical space for which every open cover has a countable subcover.

Recall that a compact space is a topological space for which every open cover has a finite subcover. Hence every compact space is a Lindelöf space.

Note: An open cover for a topological space X is a collection,

, of open sets U such that

A subcover of

for X is a subset of

which is also an open cover of X.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Ernst Lindelöf

Though all compact spaces are Lindelöf, not all Lindelöf spaces are compact. For example, the rationals

are not compact, but they are certainly Lindelöf since they are countable. In fact, any countable non-compact space is a non-compact Lindelöf space.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Ernst Lindelöf

So can you name a space that isn't Lindelof?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Ernst Lindelöf

Last edited by JaneFairfax (2008-11-24 23:42:27)