Sorgenfrey line: Difference between revisions

Latest revision as of 01:22, 28 January 2012

This article describes a standard counterexample to some plausible but false implications. In other words, it lists a pathology that may be useful to keep in mind to avoid pitfalls in proofs
View other standard counterexamples in topology

Definition

The Sorgenfrey line is defined as follows: as a set, it is the real line, and its basis of open sets is taken as all the right-open, left-closed intervals, viz., sets of the form $[a, b)$ . Equivalently, we can say that it is obtained by giving the lower limit topology corresponding to the usual ordering on the real line.

The product of two copies of the Sorgenfrey line is the Sorgenfrey plane, which is not normal. This gives an example of the fact that a product of two normal spaces need not be normal.

Topological space properties

Property	Satisfied?	Explanation	Corollary properties satisfied/dissatisfied
Separation
normal space	Yes		satisfies: completely regular space, regular space, Hausdorff space, T1 space, Kolmogorov space
hereditarily normal space	Yes		satisfies: normal space
perfectly normal space	Yes	Sorgenfrey line is perfectly normal	satisfies: perfect space, hereditarily normal space
monotonically normal space	Yes	Sorgenfrey line is monotonically normal	satisfies: hereditarily normal space
Connectedness
totally disconnected space	Yes		dissatisfies: connected space
Countability
first-countable space	Yes
separable space	Yes
second-countable space	No
Compactness
Lindelof space	Yes
paracompact Hausdorff space	Yes		satisfies:normal space
sigma-compact space	No
locally compact space	No	Sorgenfrey line is not locally compact
Miscellaneous
Baire space	Yes	Sorgenfrey line is Baire
elastic space	No	Sorgenfrey lineis not elastic	dissatisfies: metrizable space

References

Textbook references

Topology (2nd edition) by James R. Munkres^{More info}, Page 82, Chapter 2, Section 13 (the term Sorgenfrey line is not used, and the line is simply alluded to as the real numbers with the lower limit topology)

@@ Line 1: / Line 1: @@
+{{standard counterexample}}
 ==Definition==
-The '''Sorgenfrey line''' is defined as follows: as a set, it is the real line, and its basis of open sets is taken as all the half-open, half-closed sets, viz sets of the form <math>[a,b)</math>. Equivalently, we can say that it is obtained by giving the [[lower limit topology]] corresponding to the usual ordering on the real line.
+The '''Sorgenfrey line''' is defined as follows: as a set, it is the real line, and its basis of open sets is taken as all the right-open, left-closed intervals, viz., sets of the form <math>[a,b)</math>. Equivalently, we can say that it is obtained by giving the [[lower limit topology]] corresponding to the usual ordering on the real line.
+The product of two copies of the Sorgenfrey line is the [[Sorgenfrey plane]], which is ''not'' normal. This gives an example of the fact that a product of  two normal spaces need not be normal.
 ==Topological space properties==
-===Properties it does satisfy===
+{| class="sortable" border="1"
+! Property !! Satisfied? !! Explanation !! Corollary properties satisfied/dissatisfied
-* [[Totally disconnected space]]: The Sorgenfrey line is totally disconnected; given any two points, we can separate them by disjoint open sets.
+|-
-* [[First-countable space]]
+! Separation !! !! !!
-* [[Separable space]]
+|-
-* [[Lindelof space]]
+| [[satisfies property::normal space]] || Yes || || satisfies: [[satisfies property::completely regular space]], [[satisfies property::regular space]], [[satisfies property::Hausdorff space]], [[satisfies property::T1 space]], [[satisfies property::Kolmogorov space]]
-* [[Paracompact space]]
+|-
-* [[Baire space]]
+| [[satisfies property::hereditarily normal space]] || Yes || || satisfies: [[satisfies property::normal space]]
-* [[Perfectly normal space]]
+|-
+| [[satisfies property::perfectly normal space]] || Yes || [[Sorgenfrey line is perfectly normal]] || satisfies: [[satisfies property::perfect space]], [[satisfies property::hereditarily normal space]]
-===Properties it does not satisfy===
+|-
+| [[satisfies property::monotonically normal space]] || Yes || [[Sorgenfrey line is monotonically normal]] || satisfies: [[satisfies property::hereditarily normal space]]
+|-
+! Connectedness
+|-
+| [[satisfies property::totally disconnected space]] || Yes || || dissatisfies: [[dissatisfies property::connected space]]
+|-
+! Countability
+|-
+| [[satisfies property::first-countable space]] || Yes || ||
+|-
+| [[satisfies property::separable space]] || Yes || ||
+|-
+| [[dissatisfies property::second-countable space]] || No || ||
+|-
+! Compactness
+|-
+| [[satisfies property::Lindelof space]] || Yes || ||
+|-
+| [[satisfies property::paracompact Hausdorff space]] || Yes || || satisfies:[[satisfies property::normal space]]
+|-
+| [[dissatisfies property::sigma-compact space]] || No || ||
+|-
+| [[dissatisfies property::locally compact space]] || No || [[Sorgenfrey line is not locally compact]] ||
+|-
+! Miscellaneous
+|-
+| [[satisfies property::Baire space]] || Yes || [[Sorgenfrey line is Baire]] ||
+|-
+| [[dissatisfies property::elastic space]] || No || [[Sorgenfrey lineis not elastic]] || dissatisfies: [[dissatisfies property::metrizable space]]
+|}
-* [[Sigma-compact space]]
+==References==
-* [[Locally compact space]]
+===Textbook references===
-* [[Metrizable space]]
+* {{booklink-defined|Munkres}}, Page 82, Chapter 2, Section 13 (the term ''Sorgenfrey line'' is not used, and the line is simply alluded to as the real numbers with the lower limit topology)