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Jul
24
answered Can anyone provide a proof that a compact set in metric space $(X,d)$ is bounded using..
Jul
23
awarded  Pundit
Jul
23
comment Can anyone provide a proof that a compact set in metric space $(X,d)$ is bounded using..
Also, the last three are not definitions of compact.
Jul
23
comment Can anyone provide a proof that a compact set in metric space $(X,d)$ is bounded using..
Using the first definition, cover $A$ by picking $x\in A$ and then covering $A$ with $\bigcup_{n\geq 1}B_n(x)$, where $B_n(x)$ is the ball of radius $n$ centered at $x$.
Jul
16
comment An open and connected subset $U\subseteq \mathbb C$ is still connected if you remove a curve that lies entirely in $U$
The image of $f$ could be a circle of radius $1$, with $U$ a disc of radius $2$. Then $U-f([0,1])$ would not be connected.
Jul
10
reviewed Close Why does the equation $\frac{1}{\sqrt{1-x}}$ equal the following?
Jul
10
reviewed Close How can I prove that this Group is Abelian?
Jul
10
reviewed Leave Open How to get the foci / focus Hyperbola
Jul
10
reviewed Close is there an online tool for solving equation of a line?
Jul
10
reviewed Leave Open “Negative” versus “Minus”
Jul
10
reviewed Leave Open Trinomial expansion variation - generalize?
Jul
10
reviewed Close trigonometric finite series equals to polynomial function
Jul
10
answered If $3x^2$ is the derivative of $x^3$, how can $f'(x)$ be a linear map?
Jul
2
asked Prove a group is not simple, and nilpotent
Jun
24
comment Difference between 'true' and 'provable'
@user2520938 You are losing sight of the intent of what you read. The original statement is just to show that truth and proof are different. You can have some facts that you want to be true, but have no way to prove them if you don't have the right deduction system. You are correct, we can add anything we want as an axiom and consider it proven. But, if you add some statement $P$ and the negation of $P$ is provable, you tend to run into problems. Again, this points out the difference between "truth" and "proof", at least as defined in mathematical systems.
Jun
24
comment Difference between 'true' and 'provable'
@user2520938 The reason we know that $17$ is prime is because there exist rules of deduction from which we can conclude it is prime. The point is that we can state a lot of things in math, but cannot prove anything without a system of deduction. We can state "$17$ is prime", but have no way of showing it without such a system. But, either $17$ is prime or it is not, regardless of whether we can prove it. Thankfully we have a system of deduction (several really) that can show it is indeed prime.
Jun
24
answered Difference between 'true' and 'provable'
Jun
19
reviewed Leave Open Convergence radius and is a series convergent in the ends of that radius
Jun
19
revised two points with same tangent line
added 41 characters in body; edited title
Jun
18
comment The topology on $X / G$ where $G$ acts on $X$
@ThePortakal This is the Sierpinski Space: en.wikipedia.org/wiki/Sierpi%C5%84ski_space