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Article
The Finite and the Infinite
In ZF set theory finiteness classes are introduced and their stability under basic set theoretical constructions are being investigated. Typical results are:
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Hausdorffs Studien zu Fundamentalkonstruktionen der Topologie
Beim Studium abstrakter Strukturen (wie z.B. topologischer Räume, Gruppen, Moduln) spielen gewisse kanonische Konstruktionen, insbesondere die von Limiten (wie Produkten und Egalisatoren bzw. Teiloder Unterobj...
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Chapter
Hausdorffs Studien zur Dimensionstheorie
In der topologischen Dimensionstheorie beschäftigt man sich mit Dimensionsfunktionen, d. h. Abbildungen D der Klasse aller topologischen Räume in die Menge ℕ ∪ -1, +∞, wobei ℕ die Menge der nicht-negativen ganzen...
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Hausdorffs Studien über Kurven, Bögen und Peano-Kontinua
Zahlreiche Ergebnisse, insbesondere der komplexen Analysis und der Flächentheorie, legen den Gedanken nahe, das Konzept einer Kurve stelle einen Grundbegriff der Topologie dar. Leider ist dem nicht so.
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Book
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Origins
In 1904 the powder keg had been exploded through the match lighted by Zermelo.
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Elementary Observations
Experience gained presenting the contents of this paper before a learned audience discloses that the wily, attentive listener will expend more energy searching for the possible hidden presence of t...
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Chapter
Disasters with Choice
As is well known there are just as many (Lebesgue–) measurable sets as there are non–measurable ones: namely 2ℵ [where ℵ = |ℝ|]. This is peculiar since we are used to the fact that in real analysis the pathologie...
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Chapter
Beauty without Choice
It seems that the well–known arguments against the axiom of choice have been exploited until today only in a negative sense.
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Chapter
Choice Principles
We believe that the ZF–axioms describe in a correct way our intuitive contemplations concerning the notion of set. The axiom of choice (AC) is intuitively not so clear as the other ZF–axioms are, but we have lear...
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Chapter
Disasters either way
There are two kinds of truth. To the one kind belong statements so simple and clear that the opposite assertions obviously could not be defended. The other kind, the so-called “deep truths”, are st...
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Chapter
Disasters without Choice
Elementare Begriffe, wie Endlichkeit und Wohlordnung hängen jeweils vom gewählten System (Σ1oder Σ2) ab;und es ist nicht ausgeschlossen, daß dieses Abhängen von wesentlichem Charakter ist: daß eine Menge a im Sys...
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Article
Injective hulls are not natural
In a category with injective hulls and a cogenerator, the embeddings into injective hulls can never form a natural transformation, unless all objects are injective. In particular, assigning to a field its alg...
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Article
Weak Factorization Systems and Topological Functors
Weak factorization systems, important in homotopy theory, are related to injective objects in comma-categories. Our main result is that full functors and topological functors form a weak factorization system i...
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Article
Free Factorizations
Given any morphism, we construct extensions of the original category in which this morphism admits certain factorizations, in particular a (retraction, section)-factorization. To this end, we solve the word pr...
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Abelian Groups: Simultaneously Reflective and Coreflective Subcategories versus Modules
We investigate full subcategories of the category Ab of Abelian groups that are simultaneously reflective and coreflective in Ab. Such subcategories are exactly those isomorphic to categories of modules that a...
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Article
Free Adjunction of Morphisms
We develop a general setting for the treatment of extensions of categories by means of freely adjoined morphisms. To this end, we study what we call composition graphs, i.e. large graphs with a partial binary ...
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Article
Categorical Properties of Probabilistic Convergence Spaces
The purpose of this paper is to discuss some categorical properties of probabilistic convergence spaces. Its main theses are: (1) the construct P-PrTop of probabilistic pretopological spaces is the extensional...
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Categorical Topology — Its Origins, as Exemplified by the Unfolding of the Theory of Topological Reflections and Coreflections before 1971
“Man is a being, intelligent and gifted with the faculty of comprehending the abstract. Thanks to this faculty, man has conceived the ideal, and realized poesy; he has conceived the infinite, and created mathemat...
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Article
Compactness and the axiom of choice
In the absence of the axiom of choice four versions of compactness (A-, B-, C-, and D-compactness) are investigated.
