Grundzüge der Mengenlehre¶
Hausdorff, F. (1914). Grundzüge der Mengenlehre.
Cited by¶
4 citations across 4 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Completeness
- metric completeness (Cauchy sequences converge in the space), order completeness (bounded subsets have suprema in the order), logical completeness (valid formulas are provable in the system), coverage completeness (every case is handled by an explicit rule), and categorical completeness (limits and colimits of all small diagrams exist in the category) — share a common abstract pattern: each names a class of internal processes whose natural endpoints are required to lie within the system, and Hausdorff's 1914 Grundzüge der Mengenlehre gave the modern axiomatic treatment of metric spaces in which completeness via the Cauchy criterion takes its canonical form.
This sourceVeit & Comp., Leipzig. Axiomatic foundation of point-set topology via the neighbourhood-system axioms; introduces the Hausdorff (T2) separation property and treats metric spaces, completeness via the Cauchy criterion, and the foundational theorems of general topology.
- metric completeness (Cauchy sequences converge in the space), order completeness (bounded subsets have suprema in the order), logical completeness (valid formulas are provable in the system), coverage completeness (every case is handled by an explicit rule), and categorical completeness (limits and colimits of all small diagrams exist in the category) — share a common abstract pattern: each names a class of internal processes whose natural endpoints are required to lie within the system, and Hausdorff's 1914 Grundzüge der Mengenlehre gave the modern axiomatic treatment of metric spaces in which completeness via the Cauchy criterion takes its canonical form.
- Order
- 2. Relation and axiom set — the binary relation (
≤,<,⊑,≼, or domain-specific symbol) and the axioms it satisfies (reflexivity or irreflexivity; antisymmetry or asymmetry; transitivity; possibly totality, density, well-foundedness, or lattice axioms — joins and meets exist for every pair), as systematized in Hausdorff's foundational treatment of ordered sets (Hausdorff, 1914)This sourceAxiomatic foundation of point-set topology via the neighbourhood-system axioms; introduces the Hausdorff (T2) separation property and treats metric spaces, completeness, and the foundational theorems of general topology.
Supported in partVerified against the work's full text
Hausdorff's Chapter IV axiomatises ordered sets via transitive, asymmetric total ordering, plus density and well-ordering, but has no reflexive partial orders or lattices.
“wenn a vor h und h vor c steht, so soll auch a vor c stehen”
- 2. Relation and axiom set — the binary relation (
- Topology
- … whose finite intersections form a basis), a closure operator (a function $\overline{\cdot}: \mathcal{P}(X) \to \mathcal{P}(X)$ satisfying the four Kuratowski closure axioms), or a neighbourhood system (a function assigning to each point a filter of neighbourhoods satisfying the Hausdorff neighbourhood axioms
This sourceAxiomatic foundation of point-set topology via the neighbourhood-system axioms; introduces the Hausdorff (T2) separation property and treats metric spaces, completeness, and the foundational theorems of general topology.
- … whose finite intersections form a basis), a closure operator (a function $\overline{\cdot}: \mathcal{P}(X) \to \mathcal{P}(X)$ satisfying the four Kuratowski closure axioms), or a neighbourhood system (a function assigning to each point a filter of neighbourhoods satisfying the Hausdorff neighbourhood axioms
Domain-specific¶
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