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108
Topological Spaces
"... this paper. The following propositions are true: (1) Let A, B be non empty sets and R 1 , R 2 be relations between A and B. Suppose that for every element x of A and for every element y of B holds hhx; yii 2 R 1 iff hhx; ..."
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this paper. The following propositions are true: (1) Let A, B be non empty sets and R 1 , R 2 be relations between A and B. Suppose that for every element x of A and for every element y of B holds hhx; yii 2 R 1 iff hhx;
The field of complex numbers
 Journal of Formalized Mathematics
"... [14], [5], [10], and [9] provide the notation and terminology for this paper. The following propositions are true: 1. PRELIMINARIES (1) For every element z of C holds z  = z. (2) For all real numbers x1, y1, x2, y2 holds (x1 + y1i) ·(x2 + y2i) = (x1 · x2 − y1 · y2)+(x1 · y2 + x2 · y1)i. (3) ..."
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[14], [5], [10], and [9] provide the notation and terminology for this paper. The following propositions are true: 1. PRELIMINARIES (1) For every element z of C holds z  = z. (2) For all real numbers x1, y1, x2, y2 holds (x1 + y1i) ·(x2 + y2i) = (x1 · x2 − y1 · y2)+(x1 · y2 + x2 · y1)i. (3) For every real number r holds (r+0i) · i = 0+ri. (4) For every real number r holds r+0i  = r. (5) For every element z of C such that z  � = 0 holds z+0i = z z+0i · z. 2. SOME FACTS ON THE FIELD OF COMPLEX NUMBERS Let x, y be real numbers. The functor x+yiCF yielding an element of CF is defined as follows: (Def. 1) x+yiCF = x+yi. The element iCF of CF is defined as follows: (Def. 2) iCF = i. We now state several propositions: (6) iCF = 0+1i and iCF = 0+1iCF.
Separated and weakly separated subspaces of topological spaces
 Formalized Mathematics
, 1991
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Mizar: the first 30 years
 Mechanized Mathematics and Its Applications
, 2005
"... The papers were selected for presentation at workshop (with two exemptions) and publication in the journal by workshop’s program committee which consists of the following: ..."
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The papers were selected for presentation at workshop (with two exemptions) and publication in the journal by workshop’s program committee which consists of the following:
Continuity of mappings over the union of subspaces
 Journal of Formalized Mathematics
, 1992
"... 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 con ..."
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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.
Introduction to the homotopy theory
 Formalized Mathematics
, 1997
"... Summary. The paper introduces some preliminary notions concerning the homotopy theory according to [15]: paths and arcwise connected to topological spaces. The basic operations ..."
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Summary. The paper introduces some preliminary notions concerning the homotopy theory according to [15]: paths and arcwise connected to topological spaces. The basic operations
The lattice of domains of an extremally disconnected space
 Formalized Mathematics
, 1992
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The Jordan’s property for certain subsets of the plane.
 Formalized Mathematics,
, 1992
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Maximal discrete subspaces of almost discrete topological spaces
 Journal of Formalized Mathematics
, 1993
"... Summary. Let X be a topological space and let D be a subset of X. D is said to be discrete provided for every subset A of X such that A ⊆ D there is an open subset G of X such that A = D ∩ G (comp. e.g., [9]). A discrete subset M of X is said to be maximal discrete provided for every discrete subset ..."
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Summary. Let X be a topological space and let D be a subset of X. D is said to be discrete provided for every subset A of X such that A ⊆ D there is an open subset G of X such that A = D ∩ G (comp. e.g., [9]). A discrete subset M of X is said to be maximal discrete provided for every discrete subset D of X if M ⊆ D then M = D. A subspace of X is discrete (maximal discrete) iff its carrier is discrete (maximal discrete) in X. Our purpose is to list a number of properties of discrete and maximal discrete sets in Mizar formalism. In particular, we show here that if D is dense and discrete then D is maximal discrete; moreover, if D is open and maximal discrete then D is dense. We discuss also the problem of the existence of maximal discrete subsets in a topological space. To present the main results we first recall a definition of a class of topological spaces considered herein. 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. Such spaces were investigated in Mizar formalism in [6] and [7]. We show here that every almost discrete space contains a maximal discrete subspace and every such subspace is a retract of the enveloping space. Moreover, if X0 is a maximal discrete subspace of an almost discrete space X and r: X → X0 is a continuous retraction, then r −1 (x) = {x} for every point x of X belonging to X0. This fact is a specialization, in the case of almost discrete spaces, of the theorem of M.H. Stone that every topological space can be made into a T0space by suitable identification of points (see [11]).