I Understanding Dedekind Cuts: How to Recognize a Cut

  • I
  • Thread starter Thread starter mikeyBoy83
  • Start date Start date
  • Tags Tags
    Cut
mikeyBoy83
I'm trying to wrap my head around these Dedekind cuts. The definition is straightforward but I'm a little confused about the downward closure part.

##x \in Q## and ##y<x \Longrightarrow y \in Q##

Does that mean that this is not a cut because it is bounded below?

{## x \in Q : x>1 \wedge x<2 ##}

Clear this up for me please.
 
Physics news on Phys.org
mikeyBoy83 said:
I'm trying to wrap my head around these Dedekind cuts. The definition is straightforward but I'm a little confused about the downward closure part.

##x \in Q## and ##y<x \Longrightarrow y \in Q##

Does that mean that this is not a cut because it is bounded below?

{## x \in Q : x>1 \wedge x<2 ##}

Clear this up for me please.

No, it's not a cut. Another way to think of it is that a cut splits the set of all rationals into two nonempty pieces: A "left" set, L, and a "right" set, R, where
  • Every rational x is either in L or R.
  • If x is in L, and y is in R, then x &lt; y
  • The real associated with the pair L,R is the unique number r that is greater than or equal to every element of L and less than or equal to every element of R
A Dedekind cut is just such an L.
 
stevendaryl wrote:

The real associated with the pair ##L,R## is the unique number ##r## that is greater than or equal to every element of ##L## and less than or equal to every element of ##R##

Since Dedekind cuts are used to construct the reals I think it would be better to say that the real number ##r## is the cut, or preferably that ##L## as defined by OP is an extended real number.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
Back
Top