### Citations

826 |
Lattice Theory
- Birkhoff
- 1948
(Show Context)
Citation Context ...nterior operations in topological spaces and by Stone [19] in connections with the theory of Boolean algebras. Subsequently, it has been extensively investigated by several authors (see e.g. Birkhoff =-=[2]-=-, Tarski [21], MacNeille [17]). Let us recall the concepts of closed and of open domains. Let X be a topological space and let A be a subset of X. A is said to be a closed domain (or regularly closed ... |

464 | A Course in Universal Algebra.
- Burris, Sankappanavar
- 1981
(Show Context)
Citation Context ...n the Mizar formalism. To the explanation we use extracts of the articles [24] and [1], which are available in the MML. 4.1 Mizared definition of lattices We mean by a lattice (see [2], [9] and comp. =-=[4]-=-) a nonempty set L together with two binary operationssandsdefined on L satisfying the following conditions : (1) asb = bsa and asb = bsa; (2) as(bsc) = (asb)sc and as(bsc) = (asb)sc; (3) a = as(asb) ... |

177 |
Applications of the theory of Boolean rings to general topology
- Stone
- 1937
(Show Context)
Citation Context ...n of the PC Mizar System. The theory of closed and of open domains has been introduced by Kuratowski [15] in connection with the closure and the interior operations in topological spaces and by Stone =-=[19]-=- in connections with the theory of Boolean algebras. Subsequently, it has been extensively investigated by several authors (see e.g. Birkhoff [2], Tarski [21], MacNeille [17]). Let us recall the conce... |

119 | Complete lattices.
- Bancerek
- 1991
(Show Context)
Citation Context .... It should be noted that our approach has lattice--theoretic foundations. Therefore, we first present necessary parts of the theory of topological spaces ([6], [7]) and the theory of lattices ([24], =-=[1]-=-) which are available in the Main Mizar Library (MML). We next present the main result in [11] on certain characterizations of an extremally disconnected topological space by means of properties of th... |

94 | An overview of the Mizar project. - Rudnicki - 1992 |

71 |
Partially ordered sets
- MacNeille
- 1937
(Show Context)
Citation Context ...gical spaces and by Stone [19] in connections with the theory of Boolean algebras. Subsequently, it has been extensively investigated by several authors (see e.g. Birkhoff [2], Tarski [21], MacNeille =-=[17]-=-). Let us recall the concepts of closed and of open domains. Let X be a topological space and let A be a subset of X. A is said to be a closed domain (or regularly closed set) if A = Int A, and A is s... |

71 | Introduction to lattice theory. - Żukowski - 1990 |

46 |
Subsets of topological spaces.
- Wysocki, Darmochwał
- 1990
(Show Context)
Citation Context ...:: PRE TOPC:def 5 \Omega (X) n A is open; end; Next, using the Functor Definition, two functors Cl A and Int A of the type Subset of X have been introduced, the first one in [6] and the second one in =-=[7]-=-. These functors yield the closure and the interior in an arbitrary topological space (comp. [15], [13] and [8]). definition let X be TopSpace, A be Subset of X; func Cl A -? Subset of X means :: PRE ... |

33 |
Theory, with an Introduction to Descriptive Set Theory (Studies
- Kuratowski, Mostowski, et al.
- 1976
(Show Context)
Citation Context ...nd of open domains The concepts of the lattices of closed domains and of open domains of a topological space have been introduced in Mizar similarly as the lattice of domains (see [23] and comp. [2], =-=[16]-=-). For given X of the type TopSpace (more for X of a type which widens to the type TopSpace) the aggregate LattStr Closed Domains of X,Closed Domains Union X,Closed Domains Meet X AE corresponding to ... |

26 |
Sur l’opération A de l’analysis situs
- Kuratowski
- 1922
(Show Context)
Citation Context ...olves quite fast, it should be mentioned, that it is the stage of the MML that fits the 3.37 version of the PC Mizar System. The theory of closed and of open domains has been introduced by Kuratowski =-=[15]-=- in connection with the closure and the interior operations in topological spaces and by Stone [19] in connections with the theory of Boolean algebras. Subsequently, it has been extensively investigat... |

19 | The lattice of natural numbers and the sublattice of it. The set of prime numbers.
- Chmur
- 1991
(Show Context)
Citation Context ...f the type Lattice, is nonempty. definition cluster strict Lattice; end; The tree of the Mizar types defined above looks like this. ' ` i j LattStr 6 ' ` i j Lattice-like LattStr = ' ` i j Lattice In =-=[5]-=-, using the Mode Definition, for every structure L of a type which widens to the type Lattice the type SubLattice of L has been introduced. Afterwards the existential cluster for this type has been do... |

18 | The lattice of domains of an extremally disconnected space
- Karno
- 1992
(Show Context)
Citation Context ...cal domains (regular sets) in the Mizar formalism. It enveloped the lattice-theoretic foundations of the theory of domains that was given in [23] and in [12]. The aim is to present the main result in =-=[11]-=- concerning certain characterizations of an extremally disconnected topological space by means of properties of the lattice of domains and the main result in [12] concerning completeness of the lattic... |

17 | Topological spaces and continuous functions.
- Padlewska, Darmochwał
- 1990
(Show Context)
Citation Context ...f topological domains in the Mizar formalism. It should be noted that our approach has lattice--theoretic foundations. Therefore, we first present necessary parts of the theory of topological spaces (=-=[6]-=-, [7]) and the theory of lattices ([24], [1]) which are available in the Main Mizar Library (MML). We next present the main result in [11] on certain characterizations of an extremally disconnected to... |

13 | New concepts in the theory of topological space – supercondensed set, subcondensed set, and condensed set - Isomichi - 1971 |

13 |
Algebraic characterizations of special Boolean rings,
- Stone
- 1937
(Show Context)
Citation Context ... first recall a definition of a class of topological spaces which is important here. A topological space X is called extremally disconnected if for every open subset A of X the closure A is open in X =-=[20]-=- (comp. [14], [8]). In [11] the following characterization has been obtained. The lattice of all domains of a topological space X is modular if and only if X is extremally disconnected. Moreover, for ... |

4 | An Introduction to PC Mizar, Mizar Users Group, Fondation Philippele Hodey - Bonarska - 1990 |

4 |
The lattice of domains of a topological space
- Watanabe
- 1992
(Show Context)
Citation Context ...orem :: TDLAT1:41 ClosedDomainsLattice X is SubLattice of DomainsLattice X; theorem :: TDLAT1:44 OpenDomainsLattice X is SubLattice of DomainsLattice X; 5.4 Completeness of the lattices of domains In =-=[12]-=-, it is proved that the lattice of all domains of a given topological space is complete. First, the following theorem has been formulated. We give also the skeleton of the Mizar proof of it. theorem :... |

2 |
Über additive und multiplikative Mengenkörper und Mengenfunktionen
- Tarski
- 1937
(Show Context)
Citation Context ...ations in topological spaces and by Stone [19] in connections with the theory of Boolean algebras. Subsequently, it has been extensively investigated by several authors (see e.g. Birkhoff [2], Tarski =-=[21]-=-, MacNeille [17]). Let us recall the concepts of closed and of open domains. Let X be a topological space and let A be a subset of X. A is said to be a closed domain (or regularly closed set) if A = I... |

1 |
Lattice Theory, Akademie
- Gratzer
- 1978
(Show Context)
Citation Context ...s formulated in the Mizar formalism. To the explanation we use extracts of the articles [24] and [1], which are available in the MML. 4.1 Mizared definition of lattices We mean by a lattice (see [2], =-=[9]-=- and comp. [4]) a nonempty set L together with two binary operationssandsdefined on L satisfying the following conditions : (1) asb = bsa and asb = bsa; (2) as(bsc) = (asb)sc and as(bsc) = (asb)sc; (3... |

1 |
Some features of the Mizar language, (these proceedings
- Trybulec
(Show Context)
Citation Context ...sults using the Mizar formalism. The paper is written for people who are a little more familiar with the Mizar language; hence, a casual reader may have to refer first to some of the cited paper e.g. =-=[22]-=-, [3]. 2 2 Topological spaces in Mizar In this section we present all necessary facts on topological spaces formulated in the Mizar formalism. To the explanation we use extracts of the articles [6] an... |

1 |
Complete lattices, Formalized
- Bancerek
- 1991
(Show Context)
Citation Context ...ism. It should be noted that our approach has lattice{theoretic foundations. Therefore, wesrst present necessary parts of the theory of topological spaces ([6], [7]) and the theory of lattices ([24], =-=[1]-=-) which are available in the Main Mizar Library (MML). We next present the main result in [11] on certain characterizations of an extremally disconnected topological space by means of properties of th... |