Details
| Original language | English |
|---|---|
| Pages (from-to) | 945-962 |
| Number of pages | 18 |
| Journal | Topology and its applications |
| Volume | 158 |
| Issue number | 7 |
| Publication status | Published - 15 Apr 2011 |
Abstract
We show that the T1-spaces are precisely the maximal point spaces of conditionally up-complete algebraic posets with the Scott topology. Moreover, we establish an equivalence between the category of T1-spaces with a distinguished base and a certain category of so-called camps. These are conditionally up-complete, algebraic and maximized posets in which every compact element is a meet of maximal elements, and they provide essentially unique algebraic ordered models for T1-base spaces. A T1-space has a damp model (a domain model that is a camp) iff it has a base not containing any free filter base. From this, it follows that all completely metrizable spaces and, more generally, all complete Aronszajn spaces have damp models. Moreover, damp models also exist for all Stone spaces; the latter representation gives rise to an equivalence and a duality for so-called Stone base spaces, extending the classical Stone duality. Furthermore, it yields a purely order-theoretical description of clopen bases for Stone spaces and, algebraically, of finitary meet bases of Boolean lattices in terms of maximal ideals.
Keywords
- Algebraic poset, Camp, Damp, Filter-complete base, Maximal point space, Minimal base, Model, Scott topology, Stone duality, T-space, Zero-dimensional
ASJC Scopus subject areas
- Mathematics(all)
- Geometry and Topology
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In: Topology and its applications, Vol. 158, No. 7, 15.04.2011, p. 945-962.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Algebraic models for T1-spaces
AU - Erné, Marcel
PY - 2011/4/15
Y1 - 2011/4/15
N2 - We show that the T1-spaces are precisely the maximal point spaces of conditionally up-complete algebraic posets with the Scott topology. Moreover, we establish an equivalence between the category of T1-spaces with a distinguished base and a certain category of so-called camps. These are conditionally up-complete, algebraic and maximized posets in which every compact element is a meet of maximal elements, and they provide essentially unique algebraic ordered models for T1-base spaces. A T1-space has a damp model (a domain model that is a camp) iff it has a base not containing any free filter base. From this, it follows that all completely metrizable spaces and, more generally, all complete Aronszajn spaces have damp models. Moreover, damp models also exist for all Stone spaces; the latter representation gives rise to an equivalence and a duality for so-called Stone base spaces, extending the classical Stone duality. Furthermore, it yields a purely order-theoretical description of clopen bases for Stone spaces and, algebraically, of finitary meet bases of Boolean lattices in terms of maximal ideals.
AB - We show that the T1-spaces are precisely the maximal point spaces of conditionally up-complete algebraic posets with the Scott topology. Moreover, we establish an equivalence between the category of T1-spaces with a distinguished base and a certain category of so-called camps. These are conditionally up-complete, algebraic and maximized posets in which every compact element is a meet of maximal elements, and they provide essentially unique algebraic ordered models for T1-base spaces. A T1-space has a damp model (a domain model that is a camp) iff it has a base not containing any free filter base. From this, it follows that all completely metrizable spaces and, more generally, all complete Aronszajn spaces have damp models. Moreover, damp models also exist for all Stone spaces; the latter representation gives rise to an equivalence and a duality for so-called Stone base spaces, extending the classical Stone duality. Furthermore, it yields a purely order-theoretical description of clopen bases for Stone spaces and, algebraically, of finitary meet bases of Boolean lattices in terms of maximal ideals.
KW - Algebraic poset
KW - Camp
KW - Damp
KW - Filter-complete base
KW - Maximal point space
KW - Minimal base
KW - Model
KW - Scott topology
KW - Stone duality
KW - T-space
KW - Zero-dimensional
UR - http://www.scopus.com/inward/record.url?scp=79952898402&partnerID=8YFLogxK
U2 - 10.1016/j.topol.2011.01.014
DO - 10.1016/j.topol.2011.01.014
M3 - Article
AN - SCOPUS:79952898402
VL - 158
SP - 945
EP - 962
JO - Topology and its applications
JF - Topology and its applications
SN - 0166-8641
IS - 7
ER -