Category:Dedekind Cuts
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This category contains results about Dedekind Cuts.
Definitions specific to this category can be found in Definitions/Dedekind Cuts.
Let $\struct {S, \preceq}$ be a totally ordered set.
Definition 1
A Dedekind cut of $\struct {S, \preceq}$ is a non-empty proper subset $L \subsetneq S$ such that:
- $(1): \quad \forall x \in L: \forall y \in S: y \prec x \implies y \in L$ ($L$ is a lower section in $S$)
- $(2): \quad \forall x \in L: \exists y \in L: x \prec y$
Definition 2
A Dedekind cut of $\struct {S, \preceq}$ is an ordered pair $\tuple {L, R}$ such that:
- $(1): \quad \set {L, R}$ is a partition of $S$.
- $(2): \quad L$ does not have a greatest element.
- $(3): \quad \forall x \in L: \forall y \in R: x \prec y$.
Pages in category "Dedekind Cuts"
The following 5 pages are in this category, out of 5 total.