nLab Dedekind cut structure

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Definition

Given a set TT with a dense linear order <\lt, a pair of subsets (L,R)(L, R) of TT with injections i L:LTi_L:L \to T and i R:RTi_R:R \to T is a Dedekind cut structure if it comes equipped with the following structure

  • an element lLl \in L
  • an element rRr \in R
  • for every element aLa \in L and bTb \in T, a function
    c d(a,b):]b,a[{cL|i L(c)=b}c_d(a, b):]b, a[ \to \{c \in L \vert i_L(c) = b\}
  • for every element aRa \in R and bTb \in T, a function
    c u(a,b):]a,b[{cR|i R(c)=b}c_u(a, b):]a, b[ \to \{c \in R \vert i_R(c) = b\}
  • for every element aLa \in L, an element
    o d(a){bL|i L(a)<i L(b)}o_d(a) \in \{b \in L \vert i_L(a) \lt i_L(b)\}
  • for every element aRa \in R, an element
    o u(a){bR|i R(b)<i R(a)}o_u(a) \in \{b \in R \vert i_R(b) \lt i_R(a)\}
  • for every element aLa \in L and bRb \in R, an element
    t(a,b)]i L(a),i R(b)[t(a, b) \in ]i_L(a), i_R(b)[
  • for every element aTa \in T and bTb \in T, a function
    L(a,b):]a,b[({cL|i L(c)=a}{cR|i R(c)=b})L(a, b):]a,b[ \to (\{c \in L \vert i_L(c) = a\} \uplus \{c \in R \vert i_R(c) = b\})

where ]a,b[]a, b[ is the open interval bounded by aa from below and by bb from above.

See also

References

Last revised on June 10, 2022 at 10:58:11. See the history of this page for a list of all contributions to it.