How to construct an order-preserving injection $f:Q\rightarrow R$ , such that the image of $f$ is discrete subspace of $R$ (set of reals).
2026-02-22 23:11:40.1771801900
Order preserving injection $f$ from set of rationals $Q$ into $R$ with discrete image.
170 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in REAL-ANALYSIS
- how is my proof on equinumerous sets
- Finding radius of convergence $\sum _{n=0}^{}(2+(-1)^n)^nz^n$
- Optimization - If the sum of objective functions are similar, will sum of argmax's be similar
- On sufficient condition for pre-compactness "in measure"(i.e. in Young measure space)
- Justify an approximation of $\sum_{n=1}^\infty G_n/\binom{\frac{n}{2}+\frac{1}{2}}{\frac{n}{2}}$, where $G_n$ denotes the Gregory coefficients
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Is this relating to continuous functions conjecture correct?
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Absolutely continuous functions are dense in $L^1$
- A particular exercise on convergence of recursive sequence
Related Questions in ORDER-THEORY
- Some doubt about minimal antichain cover of poset.
- Partially ordered sets that has maximal element but no last element
- Ordered set and minimal element
- Order relation proof ...
- Lexicographical covering of boolean poset
- Every linearly-ordered real-parametrized family of asymptotic classes is nowhere dense?
- Is there a name for this property on a binary relation?
- Is the forgetful functor from $\mathbf{Poset}$ to $\mathbf{Set}$ represented by the object 2?
- Comparing orders induced by euclidean function and divisibility in euclidean domain
- Embedding from Rational Numbers to Ordered Field is Order Preserving
Related Questions in ORDERED-GROUPS
- Uniqueness of total orders on abelian groups
- Does every ordered divisible abelian group admit an expansion (and how many) to an ordered field?
- On totally ordered abelian groups having exactly one convex subgroup
- Definition of totally ordered monoid?
- Totally ordered abelian group with a unique " isolated subgroup"
- Continuity of absolute value in topological ordered abelian groups
- Looking at Automorphisms of Subgroups of $(\mathbb R,+)$ With Positive Slope
- Order preserving injection $f$ from set of rationals $Q$ into $R$ with discrete image.
- isomorphism in ordered monoids
- an ordered abelian group has no order units
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Enumerate $Q = \{q_n : n \in \omega \}$. We are going to define $f(q_n)$ by induction on $n$. This is done by defining an auxiliary sequence $(I_n)_{n \in \omega}$ of disjoint open intervals which do not overlap and have distinct endpoints so that $q_n \in I_n$ for every $n$.
Start by mapping $q_0$ anywhere you want and take $I_0$ to be any bounded open interval around $f(q_0)$.
Having defined $(f(q_i))_{i<n}$, map $q_n$ aproprietly so that $f(q_n) \notin \bigcup_{i<n} I_i$. Now we can still find an open interval $I_n$ around $f(q_n)$ which is disjoint from $\bigcup_{i<n} I_i$ and does not share any endpoints.
I leave the details to you.