# Tagged Questions

24 views

### Point-set topology - derived sets, boundaries, etc.

Consider the metric space $(\mathbb R^2,\sigma)$, where $\sigma$ is the discrete metric. Let $$E=\lbrace(x,y)\in \mathbb R^2 : x^2+y^2<1 \rbrace$$ i.e. the unit disc. Find the following sets ...
16 views

### Let Sn={$x\in Rn+1$; $\langle\ x,x\rangle$=1} a sphere n-dimensional. [on hold]

I study Metric Spaces and I have this problem. Let $\Bbb S^n=\{x\in \Bbb R^{n+1}; \langle x,x\rangle=1\}$ be a $n$-dimensional sphere. The projective space with dimension $n$ is the set $\Bbb P^n$, ...
30 views

### Where was I supposed to use compactness in this proof that a compact subspace can be separated from a point by open sets?

Can anyone check my proof below? P. Let $X$ be a metric space. Prove that if $K\subseteq X$ is compact and $x\notin K$, there exist disjoint open sets $U$ and $V$ such that $K\subseteq U$ and ...
39 views

### Point-set topology, characterizing sets

I think the question sort of speaks for itself. Please no solutions in the answers - I'm mostly looking to see if my logic makes sense: Let $E$ be a set in a metric space $(X,d)$. Let $E^{\circ}$ ...
16 views

I don't know if there is a name for the measure, so let me construct it: Given a metric space $X$ we define the outer measures $H_\delta^\alpha$ by $\forall A \subset X$ $H_\delta^\alpha ... 2answers 45 views ### Definition of neighborhood I am starting to work through Rudin's Principles of Mathematical Analysis. For$(X, d)$a metric space and$x \in X$, Rudin defines the neighborhood$N_r(x)$of$x$to be the set consisting of all ... 2answers 55 views ### Proof for distances to a set With a metric space$(X,d)$, prove that$|d_E(x)-d_E(y)|\leq d(x,y)+d(y,z)$. In this context,$x \in X$,$d_E(x)=\inf\left\{d(x,z) : z \in E\right\}$, E is a subset of X. I've already proved the ... 1answer 38 views ### No direct proofs of “if$ f: (X, d_X) \to (Y, d_Y)$is continuous and$X$is compact then$f$is uniformly continuous.” I am studying the theorem "if$f:(X,d_X)\to (Y,d_Y)$is continuous and$X$is compact, then$f$is uniformly continuous." I am not looking for a proof, but I have an argument against any attempt at a ... 0answers 38 views ### Showing equality of sets in$C[a,b]$The exercise states: Let$a,b\in\mathbb{R}$,$a<b$and let$(C[a,b],\Vert\cdot\Vert)$denote the vector space of continuous real functions on$[a,b]$endowed with the uniform norm. Let ... 0answers 45 views ### convergence of a sequence of cauchy I would like to ask something about the convergence of a Cauchy sequence in a space$X$metric. There will be a metric space$X$such that If$(x_n)$cauchy sequence in$X$then$(x_n)$is not ... 3answers 61 views ### The “Circle” is a Vector Space? Consider the set of angles$C = [0, \ 2\pi)$and, for all$x,y \in C$, define the$sum$operation as the sum modulo$[0, \ 2\pi)$. The identity element of the addition is the angle$0$. The inverse ... 2answers 52 views ### Some special Metric on R Apart from usual and discrete metric is there a metric on R which satisfy: d(x, y) = d(x+r , y+r) where x and y are any real no. and r is arbitary real no. Similarly is there a ... 0answers 64 views ### Condition on Equality of closure of open ball and closed ball, suppose I have a counterexample Let$(X, d)$be a metric space. Also for$x \in X$and$r \ge 0$define: $$B(x,r) = \{ y \in X : d(x,y) < r \} \quad \mbox{ and } \quad K(x,r) = \{ y \in X : d(x,y) \le r \}.$$ Denote by ... 1answer 71 views ### Continuity, and continuity in topology. Metric spaces: Neighborhood of a point$a$is a Set of point$N$, such that$\exists\delta>0:B_\delta(a)\subset N$($B_r(x)$= open ball at x of radius r) Definition of open set: "A subset$O$of ... 1answer 33 views ### Definition of a metric-nonnegativity condition There is a question in my mind which seems to be silly but I am desperately wanting the answer. Why a metric is defined from$X\times X$to$\mathbb R$and not to the set of nonnegative reals? I ... 0answers 38 views ### amazing boundedness problem from maximal function Let$n\geq 2$. For any$M>1$, prove that there exists a constant$C_M>1$such that for any ball$B$in$\mathbb{R}^n$, if we denote$MB$as the concentric ball of$B$with$M$times radius of ... 0answers 31 views ### Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces Let's$(X_1,d_1), (X_2,d_2)$be compact metric spaces such that for every finite subset of$X_1$like$A$(respectively any finite subset of$X_2$like$B$) there exists a finite subset of$X_2$... 7answers 151 views ### Convergence in a metric space Is it possible to define a metric on$\mathbb R$such that$(1,0,1,0,...)$converges on$(\mathbb R, d)$? I believe it is impossible. But how to show analytically? Any hint would be appreciated. 0answers 14 views ### semi linear uniform space In semi-linear uniform space, if$f$is a function from$(X ,Γ_X)$to$(Y,Γ_Y)$that is linear and bounded ,is$f$then continuous? Is the converse true? 0answers 60 views ###$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $In a metric space$(M,d)$the triangle inequality$d (x, z) \le d(x, y) + d (y, z)$gives us's the inequalitie $$\quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}}$$ ... 1answer 67 views ### Prove that A is both open and closed. [closed] Let$X = \{ z : | z | \leq 1 \} \cup \{ z : | z - 3 | < 1 \} $be a subset of$\mathbb{C}$. Let the metric be the usual metric$d(x,y) = | x-y |$. Prove that the set A =$\{ z : | z | \leq 1 \}$... 1answer 22 views ### Is an open connected subset of Euclidean space a countable sum of open precompact connected subsets? Let$U$be an open subset in$\mathbb R^n$. Then there exists a sequence$(U_n)_{n=1}^\infty$of open precompact subsets of$\mathbb R^n$such that$U_n \subset cl U_{n+1} \subset U$and ... 1answer 56 views ### The real numbers as a completion of the rationals The real numbers are the completion (i.e. Cauchy sequences modulo equivalence) of the rational numbers. I want to define the real numbers this way, but without using uniform spaces. The problem is ... 2answers 72 views ### Question on two metric spaces properties Question: Let$X$be a set and let$d_1$and$d_2$be two metrics on$X$. Assume that there exists a constant$C > 0$such that$d_1(x, y) \le C\, d_2(x, y)\ \ \ \forall x, y \in X$. ... 1answer 36 views ### Existence of fixed point I will copy the definition I am using just to make things clearer. Def. Let$(X,d)$be a metric space and let$F:A(\subset X)\rightarrow X$. We say F is a contraction if there exists$\lambda$where ... 1answer 43 views ### Metric space and closed sets (book misprint?) I am not sure if there is a misprint in this corollary or if I am not getting the idea right. Corollary. Let$X$be a metric space and let$A\subset X$. Then A is closed in$X$iff: $$... 2answers 43 views ### A Fundamental Property of Metric Spaces … Let (X,d) be a metric space and A\subset X and also suppose that G is open in X prove the identity:$$ \overline {G\cap A}=\overline {G\cap \overline A} $$Proposition: The intersection of ... 1answer 59 views ### Some Property of Cantor set? Draw a Cantor set C on the circle and consider the set A of all chords between points of C. Prove that A is compact. Is A convex? The proof of first part goes as follows: As we know ... 1answer 76 views ### Topology. Why is T^{-1} continuous? Today we did this proof, but we could not finish it and our prof said that the end would be easy, but I could not finish this proof. Let X be a T_3 space with a countable basis B. Then we ... 1answer 27 views ### Some Topological Properties of Starlike Sets! A subset E of \mathbb R^n is starlike if it contains a point p_0 (called a center for E) such that for each q\in E, the segment between p_0 and q lies in E. For more information please ... 2answers 144 views ### Compact metric connected space If I have a compact metric space X such that for all a,b \in X, there are points a:=x_1,...x_n=:b such that d(x_i,x_{i+1})< \varepsilon, then this space is connected. Somehow, I don't see ... 1answer 17 views ### Continous function on compact interval - bounded Let K be a compact interval in \mathbb{R}. Then every continous function \phi :K\rightarrow \mathbb{R}^d is automatically bounded. Is this a consequence of; the image of a compact is compact ? ... 1answer 30 views ### Hausdorff distance and union of sets Let X be a metric space; A_1, A_2, B_1, B_2 be non-empty subsets in X. Let d(\cdot,\cdot) be the Hausdorff distance between sets in X. Then$$ d (A_1 \cup A_2 , B_1 \cup B_2) \leq \max ... 2answers 29 views ### Something not working out for me in the continuity definition I'm studying analysis and I've ran into this proposition saying that a function from a metric space X to a metric space Y, is continuous if and only if for every open set O in Y, the inverse image of ... 1answer 21 views ### To show closedness of a subset in a metric spaces Let$(X, d)$be a metric space and$p\in X$,$\delta>0$be fixed. Let $$A=\{q \in X : p \in X, d(p,q)>\delta\}$$ How to show that$A$is closed? I tried to show that directly by taking$A$'s ... 2answers 33 views ### Lipschitz does not imply fixed points I have the following problem in mind: Let us say we have a function$f:X\rightarrow X$(X is a complete metric space) and it respects that if$x\neq y$then :$d(f(x),f(y))<d(x,y)$. My trouble ... 1answer 27 views ### Verify why this is not a metric$d(x,y)=\|x-y\|_p$($\|x\|_p=p^{-h}$if$x=p^h\dfrac{m}{n}$).$d(x,y)=\|x-y\|_pp$is prime and$\|x\|_p=p^{-h}$if$x=p^h\dfrac{m}{n}$, where$m, n$are coprimes with$p$. This is not a metric because if$x=y=p^k\dfrac{m}{n}$, then$x-y=0=p^0\dfrac0n$. ... 0answers 32 views ### Differentiation of Radon measures Assume$\ (X,d)$is a locally compact metric metric space and$\ \nu,\, \mu$are Radon measures on$X$. Then, suppose that the following hypothesis hold:$\ w\in ...
According to the definition I am using, an isometry is a mapping $f:X \rightarrow Y$ between two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$: $$d_{Y}(f(a),f(b)) = d_{X}(a,b)$$ for all $a,b \in X$ I ...