Intereting Posts

Finite ring extension and number of maximal ideals
Why is the the $k$-th derivative a symmetric multilinear map?
How to prove that if a prime divides a Fermat Number then $p=k\cdot 2^{n+2}+1$?
Show that $d \geq b+f$.
Penrose tilings as a cross section of a $5$-dimensional regular tiling
How to make sense out of the $\epsilon-\delta$ definition of a limit?
Polynomial fitting where polynomial must be monotonically increasing
If $\alpha$ is an irrational real number, why is $\alpha\mathbb{Z}+\mathbb{Z}$ dense in $\mathbb{R}$?
Finite non-abelian group with centre but no outer automorphism
How to prove by induction that $a^{2^{k-2}} \equiv 1\pmod {2^k}$ for odd $a$?
Cardinality of quotient ring $\mathbb{Z_6}/(2x+4)$
Minkowski's inequality
Show that $S$ is a group if and only if $aS=S=Sa$.
Evaluate:: $ 2 \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n+1}\left( 1 + \frac12 +\cdots + \frac 1n\right) $
Golbach's partitions: is there always one common prime in $G(n)$ and $G(n+6)$ , $n \ge 8$ (or a counterexample)?

**Let $(M,d)$ be a metric space and define: $d’ : M$x$M \rightarrow R$**

**Show that $d'(x,y)=min${$1,d(x,y)$} induces the same topology as $d$**

I know that $d’$ defines a metric on M, since d is a matric (satisfying positivity, symmetry and triangle inequality. But how do I show it induces the same topology?

- Example of a Borel set that is neither $F_\sigma$ nor $G_\delta$
- Discrete non archimedean valued field with infinite residue field
- The Cantor set is homeomorphic to infinite product of $\{0,1\}$ with itself - cylinder basis - and it topology
- Explicit homeomorphism between open and closed rational intervals?
- Decomposing a circle into similar pieces
- Show finite complement topology is, in fact, a topology

- Definition of CAT(0) metric space
- Why do we need surjectivity in this theorem?
- are two metrics with same compact sets topologically equivalent?
- Is there a topology on the full transformation semigroup?
- Measure theory and topology books that have solution manuals
- Real Analysis: open and closed sets
- Showing one point compactification is unique up to homeomorphism
- Proof that $\omega^\omega$ is completely metrizable and second countable
- Intuitive significance open sets (and software for learning topology?)
- Closed set in $\ell^1$

**Hint:** Show that:

- For any
*ball*defined with the $d$ metric, you can find a ball defined with the $d’$ metric contained in the $d$-ball. - For any $d’$-ball, show you can find a $d$-ball contained in it. You only need to work with balls since they form a basis for the metric topology.

I’ll help you do the second part, and I’ll leave the first part to you. Suppose we have $B_{d’}(x,r)$. I claim $B_{d}(x,r) \subseteq B_{d’}(x,r)$. Why? Let $y \in B_{d}(x,r)$. Then $d(x,y) < r$. But since $d’$ is the minimum of $d$ and $1$, then $d’ \leq d$. So we have $d'(x,y) \leq d(x,y) < r$, which implies $d'(x,y) < r$, so $y \in B_{d’}(x,r)$. That shows us $B_{d}(x,r) \subseteq B_{d’}(x,r)$.

**By the way**: The purpose of this assignment, and the take home lesson you should take from it once you’re done proving it, is that in the metric topology, only the small open balls matter. We can discard all larger open balls. So if you threw away all open balls of radius $3$ or larger, you would still have the same topology. If you threw away all balls of radius $.0000002$ or larger, you’d still have the same topology. Only the small balls matter.

Since we are in metric spaces, we have nice definition of closed sets, that a limit point is contained in it, so we can try approaching this problem with that fact too. Showing that they induce the same topology is equivalent to showing that any closed set in one metric is closed in the other one also (because complement of open sets is closed)

Let $A$ be a closed set in $(X,d)$ and $x_n$ be any converging sequence in $A$ such that $(x_n) \to x \in A$. Therefore, eventually $d(x_n,x)<\epsilon<1 \implies d'(x_n, x) \leq d(x_n,x)<\epsilon<1$. Hence, $(x_n) \to x \in A$ in the metric $(X,d’)$.

For the other way round, let $A$ be a closed set in $(X,d’)$ and $x_n$ be any converging sequence in $A$ such that $(x_n) \to x \in A$. Therefore, eventually $d'(x_n,x)<\epsilon<1$. Since, $d'(x,y) < 1 \implies d(x,y) = d'(x,y) \forall x, y\in X$, we have that $d(x_n, x) = d'(x_n,x)<\epsilon<1$ eventually. Hence, $(x_n) \to x \in A$ in the metric $(X,d)$.

- Problems with $\int_{1}^{\infty}\frac{\sin x}{x }dx$ convergence
- If $G/Z(G)$ is abelian then $G$ is abelian?
- Alternative ways to show that the Harmonic series diverges
- How to find intersection of two lines in 3D?
- dissection of rectangle into triangles of the same area
- Prove that $\lim_{x\to\infty}\frac{f(x)}x=\lim_{x\to\infty}f'(x)$
- $ 1^k+2^k+3^k+…+(p-1)^k $ always a multiple of $p$?
- Conjecture: Every analytic function on the closed disk is conformally a polynomial.
- A convergent-everywhere expression for $\zeta(s)$ for all $1\ne s\in\Bbb C$ with an accessible proof
- Properties of generalized limits aka nets
- For a ring $R$, does $\operatorname{End}_R(R)\cong R^{\mathrm{op}}$?
- Probability of a random $n \times n$ matrix over $\mathbb F_2$ being nonsingular
- How to solve this combinatorics problem from past TechGIG competition?
- Closed form for $(a_n)$ such that $a_{n+2} = \frac{a_{n+1}a_n}{6a_n – 9a_{n+1}}$ with $a_1=1$, $a_2=9$
- Find the area of the region which is the union of three circles