# What is Separable: Definition and 196 Discussions

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence

{

x

n

}

n
=
1

{\displaystyle \{x_{n}\}_{n=1}^{\infty }}
of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.
Like the other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset.
Contrast separability with the related notion of second countability, which is in general stronger but equivalent on the class of metrizable spaces.

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1. ### I Separable Hamiltonian for central potential

In a central potential problem we have for the Hamiltonian the expression: ##H=\frac{p^2}{2m}+V(r)## and we use to solve problems like this noting that the Hamiltonian is separable, by separable I mean that we can express the Hamiltonian as the sum of multiple parts each one commuting with the...
2. ### Solving separable 2nd order DE

This is a physics problem from Griffith's Electrodynamics. I'm mainly asking about the math here. I found the DE in the box at part (d). To solve it, I did: ##\sqrt V {d^2 V} = \beta dx^2## Integrating twice: ##\frac {4} {15} V^{2.5} = \beta x^2/2## Why is my method wrong? Thanks for the help.

Greetings, I have a question to the following section of the book https://www.springer.com/gp/book/9783319163741: I understand that the equation is separable, since I can just write $$\int_{x_0}^{x} \frac {1}{V(x', \xi, \eta)}dx' =\int_{0}^{t}dt' .$$ However, without knowing the exact shape...
4. ### I Can Non-Separable ODEs Be Solved with Coordinate Transformations?

I fell upon such an equation : $$-E'(v)a(1+\frac{cE(v)}{\sqrt{E(v)^2-1}})=vE(v)+c\sqrt{E(v)^2-1}$$ It's not separable in E on one side and v expression on the other. So I'm looking for methods to solve this maybe changes of coordinates ?
5. ### Show that the line state is separable

I introduced the unitary transformation ##U=U_a \otimes U_b## with ##(U_a\otimes 1):\;|s,s> \rightarrow\,\frac{1}{\sqrt{d}}\sum_t \omega^{ts}|t,s> ## und ##(1\otimes U_b):\;|s,s> \rightarrow\,\frac{1}{\sqrt{d}}\sum_t \omega^{-ts}|s,t> ## ##(\omega=e^{2\pi i/d}##) and let it act on the state in...
6. ### MHB -2.2.2 Separable eq y'=(x^2)/y(1+x^3)

1000 use Separable Equations to solve $$y'= \frac{x^2}{y(1+x^3)}$$ Multiply both sides by the denominator $$y(1+x^3)y'=x^2$$ Subtract $x^2$ from both sides $$-x^2 +y(1+x^3)y'=0$$ ok was trying to follow an example but ?
7. ### MHB -2.2.1 separable variables y'=\frac{x^2}{y}

2000 $\textsf{solve the given differential equation}$ $$y'=\frac{x^2}{y}$$ ok this is a new section on separable equations so i barely know anything but wanted to post the first problem hoping to understand what the book said. thanks ahead...
8. ### Solve Separable Diff. Eqn.: (y-1)dx+x(x+1)dy=0

Homework Statement ##(y-1)dx+x(x+1)dy=0## Homework EquationsThe Attempt at a Solution [/B] I multiplied both equation with, ##\frac {1} {(y-1)x(x+1)}## so I get ##\frac {dx} {x(x+1)}+\frac {dy} {y-1}=0## taking integral for both sides then I get ##ln(x)-ln(x+1)+ln(y-1)=ln(c)## so ##ln(\frac...
9. ### I Fredholm integral equation with separable kernel

Hi at all On my math methods book, i came across the following Fredholm integ eq with separable ker: 1) φ(x)-4∫sin^2xφ(t)dt = 2x-pi With integral ends(0,pi/2) I do not know how to proceed, for the solution...
10. ### MHB Separable Polynomials - Dummit and Foote - Proposition 37 .... ....

I am reading David S. Dummit and Richard M. Foote : Abstract Algebra ... I am trying to understand the proof of Proposition 37 in Section 13.5 Separable and Inseparable Extensions ...The Proposition 37 and its proof (note that the proof comes before the statement of the Proposition) read as...
11. ### I Separable Polynomials - Dummit and Foote - Proposition 37

I am reading David S. Dummit and Richard M. Foote : Abstract Algebra ... I am trying to understand the proof of Proposition 37 in Section 13.5 Separable and Inseparable Extensions ...The Proposition 37 and its proof (note that the proof comes before the statement of the Proposition) read as...
12. ### MHB Separable Polynomials - Remarks by Dummit and Foote .... ....

Dummit and Foote in Section 13.5 on separable extensions make some remarks about separable polynomials that I do not quite follow. The remarks follow Corollary 34 and its proof ... Corollary 34, its proof and the remarks read as follows: https://www.physicsforums.com/attachments/6639 In the...
13. ### I Separable Polynomials - Remarks by Dummit and Foote .... ....

Dummit and Foote in Section 13.5 on separable extensions make some remarks about separable polynomials that I do not quite follow. The remarks follow Corollary 34 and its proof ... Corollary 34, its proof and the remarks read as follows: In the above text by D&F, in the remarks after the...
14. ### MHB Separable Polynomials - Paul E Bland's definition and example ....

I am reading Paul E Bland's book: The Basics of Abstract Algebra and I am trying to understand his definition of "separable polynomial" and his second example ... Bland defines a separable polynomial as follows:https://www.physicsforums.com/attachments/6636... and Bland's second example is as...
15. ### I Separable Polynomials - Paul E Bland's definition and exampl

I am reading Paul E Bland's book: The Basics of Abstract Algebra and I am trying to understand his definition of "separable polynomial" and his second example ... Bland defines a separable polynomial as follows: ... and Bland's second example is as follows: I am uncomfortable with, and do...
16. ### MHB Splitting Fields and Separable Polynomials ....

I am reading both David S. Dummit and Richard M. Foote : Abstract Algebra and Paul E. Bland's book: The Basics of Abstract Algebra ... ... I am trying to understand separable polynomials ... ... but D&F and Bland seem to define them slightly differently and interpret the application of the...
17. ### I Splitting Fields and Separable Polynomials ....

I am reading both David S. Dummit and Richard M. Foote : Abstract Algebra and Paul E. Bland's book: The Basics of Abstract Algebra ... ... I am trying to understand separable polynomials ... ... but D&F and Bland seem to define them slightly differently and interpret the application of the...
18. ### Stuck on separable equation relating to moment of inertia

Homework Statement (a) Consider a cylindrical can of gas with radius R and height H rotating about its longitudinal axis. The rotation causes the density of the gas, η, to obey the differential equation dη(ρ)/dp = κ ω2 ρ η(ρ) where ρ is the distance from the longitudinal axis, the constant κ...
19. ### MHB The extension KE/F is finite and separable

Hey! Let $C$ be an algebraic closure of $F$ and let $f\in F[x]$ be separable. Let $K\leq C$ be the splitting field of $f$ over $F$ and let $E\leq C$ be a finite and separable extension of $F$. I want to show that the extension $KE/F$ is finite and separable. We have that $KE$ is the smallest...
20. ### I Understanding Separable Vector Spaces: The Basics Explained

Dear forum, I am trying to understand what a separable vector space is. I know we can perform the tensor product of two or more vector space and obtain a new vector space. Is that vector space separable because it is the product of other vector spaces? thanks
21. ### First order separable Equation ODE

Homework Statement \frac{dy}{dx}\:+\:ycosx\:=\:5cosx I get two solutions for y however only one of them is correct according to my online homework (see attempt at solution) Homework Equations y(0) = 7 is initial condition The Attempt at a Solution \int \:\frac{1}{5-y}dy\:=\:\int...
22. ### MHB Why is this polynomial separable?

Hey! :o In my notes there is the following: Let $F$ be a field. The irresducible $f\in F[x]$ is separable, if all the roots are different. A non-constant polynomial $f\in K[x]$ is separable, if all the irreducible factors are separable. Example: $f(x)=(x^2-2)^2(x^2+3)\in \mathbb{Q}[x]$...
23. ### MHB The extension is Galois iff E is a splitting field of a separable polynomial of F[x]

Hey! :o Let $E/F$ be a finite extension. I want to show that this extension is Galois if and only if $E$ is a splitting field of a separable polynomial of $F[x]$. I have done the folllowing: $\Rightarrow$ : We suppose that $E/F$ is Galois. So, we have that the extension is normal and...
24. ### I Constant solution and uniqueness of separable differential eq

Hi, I am learning ODE and I have some problems that confuse me. In the textbook I am reading, it explains that if we have a separable ODE: ##x'=h(t)g(x(t))## then ##x=k## is the only constant solution iff ##x## is a root of ##g##. Moreover, it says "all other non-constant solutions are separated...
25. ### MHB The irreducible polynomial is not separable

Hey! :o Let $F$ be a field, $D=F[t]$, the polynomial ring of $t$, with coefficients from $F$ and $K=F(t)$ the field of rational functions of $t$. (a) Show that $t\in D$ is a prime element of $D$. (b) Show that the polynomial $x^n-t\in K[x]$ is irreducible. (c) Let $\text{char} F=p$. Show...
26. ### I Interval of existence and uniqueness of a separable 1st ODE

Problem: y'=((x-1)/(x^2))*(y^2) , y(1)=1 . Find solutions satisfying the initial condition, and determine the intervals where they exist and where they are unique. Attempt at solution: Let f(x,y)=((x-1)/(x^2))*(y^2), which is continuous near any (x0,y0) provided x0≠0 so a solution with y(x0)=y0...
27. ### Separable Differential Equation

Homework Statement Solve the differential equation: (ex+1)cosy dy + ex(siny +1)dx=0 y(0)=3 Homework Equations none The Attempt at a Solution (ex+1)cosy dy + ex(siny +1)dx=0 (ex+1)cosy dy =- ex(siny +1)dx cosy/(siny+1)dy=-ex/(ex+1)dx ∫cosy/(siny+1)dy=-∫ex/(ex+1)dx using u sub on both the...
28. ### Separable differential equation

Homework Statement Solve each of the following differential equations: 4xydx + (x2 +1)dy=0Homework Equations None The Attempt at a Solution 4xydx + (x2 +1)dy=0 (x2 +1)dy=-4xydx dy/y=-(4xdx)/(x2 +1) ∫dy/y=∫-(4xdx)/(x2 +1) ln|y|=-2ln|x2+1| +C used u-sub on last step fo u=x2 +1
29. ### MHB -10.5.99 separable eq y'=x^4y^4

Solve the separable differential equation $\displaystyle y'=x^4y^4$ Solve for $y$ if possible. $\displaystyle y=\frac{{y'}^{(1/4)}}{x}$ Not sure ?
30. ### Difficult Separable Integration Problem

Homework Statement Q=-1*K(T)*(H*W)*(dT/dx)+((I^2)(p)(dx)/(H*W)) K(T)=(197.29-.06333333(T+273)) H=0.01905 W=0.06604 I=700 p=10*10^-6 Q=some constant Please separate and differentiate to solve for Q using variables of T and x. Boundaries: T: Upper=T1 (constant) Lower=T0 (constant) x: Upper=L...
31. ### MHB Help with another separable equation

I am really struggling with this one, if anyone can help. (ln(y))3*(dy/dx)=(x^3)y with initial conditions y=e^2 x=1 I get c=4/(e^4) - 1/4 then I get stuck at (3ln^2y - ln^3y)/(y^2)=(x^4)/4 + C Any ideas? I'm really not good at these so there are probably mistakes, because at this point I have...
32. ### MHB How to Solve a Separable Equation with Initial Condition u(0)=6?

4 du/dt = u^2 with initial condition u(0)=6 I have worked this multiple times, and all I get is u = (-8/(t-27))^(1/3) and it is NOT right! If anyone can help it would be very appreciated.
33. ### Can V and x be Separated in this Differential Equation?

Homework Statement i am asked to form a differential equation using dy/dx = 1 + y + (x^2 ) + y(x^2) , but i gt stucked here , homework to proceed? as we can see , the V and x are not separable Homework EquationsThe Attempt at a Solution
34. ### Whittaker's solution and separable variables

So It is well known that the 2D solution to the Laplace equation can be obtained by changing to complex coordinates ##u=x+iy## and ##v=x-iy##. This can be extended to n dimensions as long as the complex coordinates chosen also solve the Laplace equation. For example in 3D...
35. ### How to prove the following defined metric space is separable

Let ##\mathbb{X}## be the set of all sequences in ##\mathbb{R}## that converge to ##0##. For any sequences ##\{x_n\},\{y_n\}\in\mathbb{X}##, define the metric ##d(\{x_n\},\{y_n\})=\sup_{n}{|x_n−y_n|}##. Show the metric space ##(\mathbb{X},d)## is separable. I understand that I perhaps need to...
36. ### Non-Separable Couples: Bell's Theorem Unites

do you think of Bell's theorem when you think of forming a couple : that it builds a unity and is not separable ?
37. ### Does local realism imply separability?

Hi. Bell's formulation of local realism is $$P(a,b)=\int\ d\lambda\cdot\rho(\lambda)p_A(a,\lambda)p_B(b,\lambda)\enspace.$$ Let's for simplicity assume there's only a finite number of states, so this becomes $$P(a,b)=\sum_{i} p_i\cdot\ p_A(a,i)p_B(b,i)\enspace.$$ I'm trying to translate this...
38. ### Help in checking the solution of this separable equation

Homework Statement It is just an evaluation problem which looks like this dx/dy = x^2 y^2 / 1+x Homework Equations dx/dy = x^2 y^2 / 1 + x The Attempt at a Solution What i did is cross multiply to get this equation y^2 dy = x^2 / 1+x dx then next line ∫y^2 dy = ∫x^2/1+x dx y^3/3 = ∫dx + ∫1/x...
39. ### First Order D.E (Not Linear, Exact, or Separable)

Homework Statement What is the general solution of: y'=(3*y^2-x^2)/(2*x-y) Homework EquationsThe Attempt at a Solution This First Order equation is neither linear nor separable. I also have checked the Exact test, which turns to be Not Exact. Any help regarding how...
40. ### Differential Equations, Separable, Simplification of answer

Homework Statement I believe I have solved this differential equation, yet do not know how the book arrived at it's answer... Solve the differential equation in its explicit solution form. The answer the book gives is... Homework Equations Separable Differential Equation The Attempt...
41. ### Differential Equations, Separable, Explicit Solution

Homework Statement Solve the differential equation, explicitly. dy/dx = (2x)/(1+2y) The answer given by the book is... -1/2 + 1/2sqrt(2x - 2x^2 +4) Homework Equations Process for solving separable differential equations The Attempt at a Solution dy/dx = (2x)/(1+2y) (1 + 2y)*dy = 2x*dx...
42. ### Proving that Every Closed Set in Separable Metric Space is Union of Perfect and Countable Set

Homework Statement Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable. (Rudin: Principles of Mathematical Analysis, 2nd ed.) Homework Equations Every separable metric space has a countable base. The...
43. ### Separable differential equations

Homework Statement [/B]Homework Equations The Attempt at a Solution I've highlighted two equations on the screenshot. How did it proceed from the first to the second? I'm actually confused with the absolute values. What is the idea behind getting rid of the first absolute value(1-5v^2) while...
44. ### MHB Why is this equation non separable?

Hi everyone, I am trying to find any particular solution for the equation dy/dx + y = 1. I have been told it is not separable. I have done the following: dy/dx = 1-y integral of 1/(1-y) dy = integral -loge(1-y) = c e^-c = 1-y y = 1- e^-c let c = 0 y = 1-1 A particular solution is y= 0. My...
45. ### Separable partial differential equation

Homework Statement I have two equations. cos(θ)wφ + sin(θ)wφ = 0 (1) And ## \frac{w_φ}{r}## + ∂wφ/∂r = 0 (2) Find wφ, which is a function of both r and theta. Homework EquationsThe Attempt at a Solution I end up with two equations, having integrated. wφ=## \frac{A}{sinθ}## from (1)...
46. ### Absolute Values in Separable Differential Equations

When solving a separable differential equation, my textbook says this: ln|v-49|=-t/5+C→ |v-49|=e-t/5+C→ v=49+ce-t/5 What happened to the absolute values? I think it has something to do with the exponential always being positive.
47. ### Why are separable spaces called "separable"?

What is getting separated from what? I presume there is some historical founding case that involved separating something. Like how the original vector spaces were mental arrows in R^3.
48. ### MHB Solving the separable equation, putting it in explicit form

Find the solution of the given initial value problem in explicit form. Determine interval which solution is defined. (which i think is the same thing as saying find the interval of validity) $y' = (1-2x)y^2$ , $y(0) = -1/6$ So here is what I have so far.. $\int y^{-2}dy = x - x^2 + C$ \$=...
49. ### MHB Solution of Separable Equation, Plotting Graph, Interval Estimation

(a) Find the solution of the given initial value problem in explicit form. (b) Plot the graph of the solution (c) Determine (at least approximately) the interval in which the solution is defined \frac{dr}{dx} =\frac{r^2}{x} and r(1) = 2 I'm kind of confused..How do I start this problem?
50. ### Solving Separable ODEs: How to Integrate with Functions of t?

I understand how to integrate this: ∫y2dy. I don't understand how to integrate this: di(t)/dt = i(t)p(t) intergrate((di(t)/dt/i(t))*dt = p(t)dt) (see this image: http://i.imgur.com/OdKI309.png) how do you perform the intergral on the left, seeing as as it not dt, but di(t)? thanks