# Sorgenfrey line

## Primary tabs

Defines:
lower limit topology
Synonym:
Sorgenfrey topology
Type of Math Object:
Example
Major Section:
Reference

## Mathematics Subject Classification

55-00 General reference works (handbooks, dictionaries, bibliographies, etc.)
54-00 General reference works (handbooks, dictionaries, bibliographies, etc.)
22-00 General reference works (handbooks, dictionaries, bibliographies, etc.)

### is the sorgenfrey line compact?

Is the sorengfrey line compact?

### Re: is the sorgenfrey line compact?

No. The simplest way to see it is to take the cover
{ [n,n+1[ | n integer }. All the sets are open and if you remove
any one of them, it is no longer a cover. Hence it cannot have
a finite subcover and so cannot be compact.

### Is The Sorgenfrey topology a Baire space?

I know that the Sorgenfrey topology is totally disconnected, but I cant seem to prove that this implies it is a baire space. I have given up trying to prove directly that it is a baire space, I couldn't get anywhere with that.

### Re: Is The Sorgenfrey topology a Baire space?

Not every totally disconnected space is a Baire space: take the rationals, for example.

But the Sorgenfrey line is indeed a Baire space, and the proof is the same as for the (Euclidean) real line.

### compact subsets of the Sorgenfrey line

Why any compact subset of sorgenfrey line must be a countable set?