Details
| Original language | English |
|---|---|
| Pages (from-to) | 217-243 |
| Number of pages | 27 |
| Journal | Applied categorical structures |
| Volume | 9 |
| Issue number | 3 |
| Publication status | Published - May 2001 |
Abstract
The core of a point in a topological space is the intersection of its neighborhoods. We construct certain completions and compactifications for densely core-generated spaces, i.e., T0-spaces having a subbasis of open cores such that the points with open cores are dense in the associated patch space. All T0-spaces with a minimal basis are in that class. Densely core-generated spaces admit not only a coarsest quasi-uniformity (the unique totally bounded transitive compatible quasi-uniformity), but also a purely order-theoretical description by means of their specialization order and a suitable join-dense subset (join-basis). It turns out that the underlying ordered sets of the completions and compactifications obtained are, up to isomorphism, certain ideal completions of the join-basis. The topology of the resulting completion or compactification is the Lawson topology or the Scott topology, or a slight modification of these.
Keywords
- (Ordered) topological space, (Quasi-)uniform space, (Strongly) sober, Cauchy filter, Compactification, Completion, Core, Ideal, Totally (order-) separated
ASJC Scopus subject areas
- Mathematics(all)
- Theoretical Computer Science
- Computer Science(all)
- General Computer Science
Cite this
- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS
In: Applied categorical structures, Vol. 9, No. 3, 05.2001, p. 217-243.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Ideal completions and compactifications
AU - Erné, Marcel
PY - 2001/5
Y1 - 2001/5
N2 - The core of a point in a topological space is the intersection of its neighborhoods. We construct certain completions and compactifications for densely core-generated spaces, i.e., T0-spaces having a subbasis of open cores such that the points with open cores are dense in the associated patch space. All T0-spaces with a minimal basis are in that class. Densely core-generated spaces admit not only a coarsest quasi-uniformity (the unique totally bounded transitive compatible quasi-uniformity), but also a purely order-theoretical description by means of their specialization order and a suitable join-dense subset (join-basis). It turns out that the underlying ordered sets of the completions and compactifications obtained are, up to isomorphism, certain ideal completions of the join-basis. The topology of the resulting completion or compactification is the Lawson topology or the Scott topology, or a slight modification of these.
AB - The core of a point in a topological space is the intersection of its neighborhoods. We construct certain completions and compactifications for densely core-generated spaces, i.e., T0-spaces having a subbasis of open cores such that the points with open cores are dense in the associated patch space. All T0-spaces with a minimal basis are in that class. Densely core-generated spaces admit not only a coarsest quasi-uniformity (the unique totally bounded transitive compatible quasi-uniformity), but also a purely order-theoretical description by means of their specialization order and a suitable join-dense subset (join-basis). It turns out that the underlying ordered sets of the completions and compactifications obtained are, up to isomorphism, certain ideal completions of the join-basis. The topology of the resulting completion or compactification is the Lawson topology or the Scott topology, or a slight modification of these.
KW - (Ordered) topological space
KW - (Quasi-)uniform space
KW - (Strongly) sober
KW - Cauchy filter
KW - Compactification
KW - Completion
KW - Core
KW - Ideal
KW - Totally (order-) separated
UR - http://www.scopus.com/inward/record.url?scp=0035331085&partnerID=8YFLogxK
U2 - 10.1023/A:1011260817824
DO - 10.1023/A:1011260817824
M3 - Article
AN - SCOPUS:0035331085
VL - 9
SP - 217
EP - 243
JO - Applied categorical structures
JF - Applied categorical structures
SN - 0927-2852
IS - 3
ER -