The Definition of Topological Manifolds

This article introduces the definition of n-locally Euclidean topological spaces and topological manifolds [13].Riccardi Marco - Via del Pero 102, 54038 Montignoso, ItalyGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383-386, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Adam Grabowski. Properties of the product of compact topological spaces. Formalized Mathematics, 8(1):55-59, 1999.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Zbigniew Karno. Separated and weakly separated subspaces of topological spaces. Formalized Mathematics, 2(5):665-674, 1991.Zbigniew Karno. The lattice of domains of an extremally disconnected space. Formalized Mathematics, 3(2):143-149, 1992.Artur Korniłowicz and Yasunari Shidama. Intersections of intervals and balls in En/T. Formalized Mathematics, 12(3):301-306, 2004.John M. Lee. Introduction to Topological Manifolds. Springer-Verlag, New York Berlin Heidelberg, 2000.Robert Milewski. Bases of continuous lattices. Formalized Mathematics, 7(2):285-294, 1998.Beata Padlewska. Locally connected spaces. Formalized Mathematics, 2(1):93-96, 1991.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Karol Pąk. Basic properties of metrizable topological spaces. Formalized Mathematics, 17(3):201-205, 2009, doi: 10.2478/v10037-009-0024-8.Bartłomiej Skorulski. First-countable, sequential, and Frechet spaces. Formalized Mathematics, 7(1):81-86, 1998.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990

Maximal anti-discrete subspaces of topological spaces

Summary. Let X be a topological space and let A be a subset of X. A is said to be anti-discrete provided for every open subset G of X either A ∩ G = /0 or A ⊆ G; equivalently, for every closed subset F of X either A ∩ F = /0 or A ⊆ F. An anti-discrete subset M of X is said to be maximal anti-discrete provided for every anti-discrete subset A of X if M ⊆ A then M = A. A subspace of X is maximal anti-discrete iff its carrier is maximal anti-discrete in X. The purpose is to list a few properties of maximal anti-discrete sets and subspaces in Mizar formalism. It is shown that every x ∈ X is contained in a unique maximal anti-discrete subset M(x) of X, denoted in the text by MaxADSet(x). Such subset can be defined by M(x) = � {S ⊆ X: x ∈ S, and S is open or closed in X}. It has the following remarkable properties: (1) y ∈ M(x) iff M(y) = M(x), (2) either M(x)

Maximal Kolmogorov Subspaces of a Topological Space as Stone Retracts of the Ambient Space 1

Summary. Let X be a topological space. X is said to be T0-space (or Kolmogorov space) provided for every pair of distinct points x, y ∈ X there exists an open subset of X containing exactly one of these points (see [1], [9], [5]). Such spaces and subspaces were investigated in Mizar formalism in [8]. A Kolmogorov subspace X0 of a topological space X is said to be maximal provided for every Kolmogorov subspace Y of X if X0 is subspace of Y then the topological structures of Y and X0 are the same. M.H. Stone proved in [11] that every topological space can be made into a Kolmogorov space by identifying points with the same closure (see also [12]). The purpose is to generalize the Stone result, using Mizar System. It is shown here that: (1) in every topological space X there exists a maximal Kolmogorov subspace X0 of X, and (2) every maximal Kolmogorov subspace X0 of X is a continuous retract of X. Moreover, if r: X → X0 is a continuous retraction of X onto a maximal Kolmogorov subspace X0 of X, then r −1 (x) = MaxADSet(x) for any point x of X belonging to X0, where MaxADSet(x) is a unique maximal anti-discrete subset of X containing x (see [7] for the precise definition of the set MaxADSet(x)). The retraction r from the last theorem is defined uniquely, and it is denoted in the text by “Stoneretraction”. It has the following two remarkable properties: r is open, i.e., sends open sets in X to open sets in X0, and r is closed, i.e., sends closed sets in X to closed sets in X0. These results may be obtained by the methods described by R.H. Warren in [16]

Université Catholique de Louvain Continuity of Mappings over the Union of Subspaces

Summary. Let X and Y be topological spaces and let X1 and X2 be subspaces of X. Let f: X1 ∪ X2 → Y be a mapping defined on the union of X1 and X2 such that the restriction mappings f|X1 and f|X2 are continuous. It is well known that if X1 and X2 are both open (closed) subspaces of X, then f is continuous (see e.g. [6, p.106]). The aim is to show, using Mizar System, the following theorem (see Section 5): If X1 and X2 are weakly separated, then f is continuous (compare also [15, p.358] for related results). This theorem generalizes the preceding one because if X1 and X2 are both open (closed), then these subspaces are weakly separated (see [5]). However, the following problem remains open. Problem 1. Characterize the class of pairs of subspaces X1 and X2 of a topological space X such that (∗) for any topological space Y and for any mapping f: X1 ∪ X2 → Y, f is continuous if the restrictions f|X1 and f|X2 are continuous. In some special case we have the following characterization: X1 and X2 are separated iff X1 misses X2 and the condition (∗) is fulfilled. In connection with this fact we hope that the following specification of the preceding problem has an affirmative answer. Problem 2. Suppose the condition (∗) is fulfilled. Must X1 and X2 be weakly separated? Note that in the last section the concept of the union of two mappings is introduced and studied. In particular, all results presented above are reformulated using this notion. In the remaining sections we introduce concepts needed for the formulation and the proof of theorems on properties of continuous mappings, restriction mappings and modifications of the topology

On discrete and almost discrete topological spaces

Summary. A topological space X is called almost discrete if every open subset of X is closed; equivalently, if every closed subset of X is open (comp. [6],[7]). Almost discrete spaces were investigated in Mizar formalism in [4]. We present here a few properties of such spaces supplementary to those given in [4]. Most interesting is the following characterization: A topological space X is almost discrete iff every nonempty subset of X is not nowhere dense. Hence, X is non almost discrete iff there is an everywhere dense subset of X different from the carrier of X. We have an analogous characterization of discrete spaces: A topological space X is discrete iff every nonempty subset of X is not boundary. Hence, X is non discrete iff there is a dense subset of X different from the carrier of X. It is well known that the class of all almost discrete spaces contains both the class of discrete spaces and the class of anti-discrete spaces (see e.g., [4]). Observations presented here show that the class of all almost discrete non discrete spaces is not contained in the class of anti-discrete spaces and the class of all almost discrete non anti-discrete spaces is not contained in the class of discrete spaces. Moreover, the class of almost discrete non discrete non anti-discrete spaces is nonempty. To analyse these interdependencies we use various examples of topological spaces constructed here in Mizar formalism

Warsaw University- Bia̷lystok On Kolmogorov Topological Spaces 1

Summary. Let X be a topological space. X is said to be T0-space (or Kolmogorov space) provided for every pair of distinct points x, y ∈ X there exists an open subset of X containing exactly one of these points; equivalently, for every pair of distinct points x, y ∈ X there exists a closed subset of X containing exactly one of these points (see [1], [6], [2]). The purpose is to list some of the standard facts on Kolmogorov spaces, using Mizar formalism. As a sample we formulate the following characteristics of such spaces: X is a Kolmogorov space iff for every pair of distinct points x, y ∈ X the closures {x} and {y} are distinct. There is also reviewed analogous facts on Kolmogorov subspaces of topological spaces. In the presented approach T0-subsets are introduced and some of their properties developed

Continuity of mappings over the union of subspaces

Summary. Let X and Y be topological spaces and let X1 and X2 be subspaces of X. Let f: X1 ∪ X2 → Y be a mapping defined on the union of X1 and X2 such that the restriction mappings f |X1 and f |X2 are continuous. It is well known that if X1 and X2 are both open (closed) subspaces of X, then f is continuous (see e.g. [7, p.106]). The aim is to show, using Mizar System, the following theorem (see Section 5): If X1 and X2 are weakly separated, then f is continuous (compare also [14, p.358] for related results). This theorem generalizes the preceding one because if X1 and X2 are both open (closed), then these subspaces are weakly separated (see [6]). However, the following problem remains open. Problem 1. Characterize the class of pairs of subspaces X1 and X2 of a topological space X such that (∗) for any topological space Y and for any mapping f: X1 ∪ X2 → Y, f is continuous if the restrictions f |X1 and f |X2 are continuous. In some special case we have the following characterization: X1 and X2 are separated iff X1 misses X2 and the condition (∗) is fulfilled. In connection with this fact we hope that the following specification of the preceding problem has an affirmative answer. Problem 2. Suppose the condition (∗) is fulfilled. Must X1 and X2 be weakly separated Note that in the last section the concept of the union of two mappings is introduced and studied. In particular, all results presented above are reformulated using this notion. In the remaining sections we introduce concepts needed for the formulation and the proof of theorems on properties of continuous mappings, restriction mappings and modifications of the topology