# Tagged Questions

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### Question regarding a wording of an exercise related to Noetherian topological space

The exercise states "If $X$ is a Noetherian topological space, show that the union of any subset of the connected components of $X$ is always open and closed in $X$." Does the question mean "If I ...
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### Does an lp norm induce a ball topology? [closed]

Namely, does the metric $$||x - y||_p$$ induce the usual ball topology that a metric induces? I wasn't able to find any results regarding this on a quick Google search.
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### Homeomorphism between positive rationals and non-negative rationals [duplicate]

Professor told me the two spaces are homeomorphic if we consider them as subspaces of R equipped with the standard topology. It's a bit surprising for me since I have already found Q is homeomorphic ...
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### Density and convergence

I have a small question: Is it true that if the basis of a space $A$ is dense in a space $B$ ($B\subset A$) then if $u_n\rightarrow u$ in $A$ we have that $u_n\rightarrow u$ in $B$ ?
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### For compact $K$ and open $U \supseteq K$, there exists $\varepsilon>0$ such that $B(K,\varepsilon) \subseteq U$

Let $X$ be a metric space. Let $K$ be a compact subset of $X$ and $U$ an open subset of $X$ containing $K$. I strongly believe and want to prove that there exists $\varepsilon>0$ such that ...
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### Deck transformation acting properly discontinuously assumed covering space is path-connected

Let $p:E\rightarrow X$ be a covering space, $x\in X$ fixed point, $E$ path-connected and $\Delta(p)$ – Deck transformation group of $p$, that is $\Delta(p) = \{f\in \text{Homeo}(E):pf=p\}$. Let ...
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### Is continuous map from covering space to itself homeomorphism assumed both cover and base path-connected and $pf=p$?

In my topology assignment I came across the following problem: True or false? Let $E$ and $X$ be path-connected. For every covering map $p:E\rightarrow X$ and continuous map $f:E\rightarrow E$ ...
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### Compact $\omega$-limit set $\Rightarrow$ connected

Consider the flow $\varphi: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ and $L_{\omega}(x)$ the $\omega$-limit set of a point $x \in \mathbb{R}^n$. How can I show that if $L_{\omega}(x)$ is ...
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### proper map between metric spaces

prove that the continues function $f:X\rightarrow Y$ is proper (preimage of a compact is compact)when $X,Y$ are metric spaces,iff for any metric space $Z$ a map from $X\times Z$ to $Y\times Z$ that ...
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Let $\mathbb{R}^2$ with the usual topology and let $$Y = A_0 \cup (\bigcup_{n \in \mathbb{N}} A_n) \cup (\bigcup_{n \in \mathbb{N}}L_n)$$ where $$A_0 = \{ 0 \} \times [0,1] \qquad A_n = \{ ... 1answer 39 views ### a topological space is the union of its irreducible components if we define irreducible component of a topological space X as the maximal closed irreducible subset of X,prove that we can write X as the union of its irreducible components. how to approach ... 1answer 40 views ### a quotient space homeomorphic with \mathbb{R}\mathbb{P^2} prove that if we glue the closed unit disc to the circular boundary of the mobius strip we obtain a quotient space that is homeomorphic with \mathbb{R}\mathbb{P^2}. it is my general topology ... 1answer 50 views ### \mathbb{R}\mathbb{P^2} as a quotient space If we construct \mathbb{R}\mathbb{P^2} by gluing the sides of I^2, can anyone explain for me that why \mathbb{R}\mathbb{P^2} can be considered as the quotient space induced by the equivalence ... 0answers 44 views ### one point compactification of upper half plane "prove that the closed unit disc could be considered as the one point compactification of the upper half plane with the X-axis as the bounday" i believe in this statement by the imagination,but i ... 1answer 18 views ### Let K_1(0) := \{x \in \mathbb{R}^2: \|x\|_2 < 1\} and S := K_1(0) \setminus \mathbb{Q}^2. Is M path connected? The Assignment: Let K_1(0) := \{x \in \mathbb{R}^2: \|x\|_2 < 1\} and S := K_1(0) \setminus \mathbb{Q}^2. Is S path connected? Explain your answer. I don't think S is path-connected since ... 2answers 26 views ### Question about convex set? Can someone prove or give me a counterexample that is for a set A, if the closure of A is convex then the interior of A is also convex? 2answers 99 views ### f:X\rightarrow S^1 a continuous map. X a path-connected topological space. Let f:X\rightarrow S^1 be a continuous map from a path-connected topological space X and let p:\mathbb{R} \rightarrow S^1 be the universal covering. What is the condition when f admits a ... 1answer 28 views ### Topology - A continuous map from the circle that can be extended to a continuous map from the disc is null a question from my h.w. Let h:S^1 \rightarrow X be continuous map, and suppose it can be extended continuously to the Disc meaning there exists a continuous map k:D^2 \rightarrow X such that ... 0answers 58 views ### Solution check please: Basic topology (open/closed sets and balls) This is a part of a problem from a past exam so i need my solution to be reviewed. Thank you in advance for taking your time :) Let \Omega be a bounded open set and let A= \{r>0| \exists z \in ... 1answer 57 views ### Subset of infinite connected set How to proove that infinite connected set has got proper infinite connected subset? 2answers 76 views ### (Certain) colimit and product in category of topological spaces Consider the diagram$$(*):\;\;\;X_0 \stackrel{i_0}\hookrightarrow X_1 \stackrel{i_1}\hookrightarrow X_2 \stackrel{i_2}\hookrightarrow \cdots $$in category of topological spaces. Denote I the ... 0answers 26 views ### Principal order filters on a POSET I have another problem, but in this one I have no idea how to start. Let be (X,\leq) a POSET with a first element and gifted with the topology \{ (a,\rightarrow) : a \in X \} (principal order ... 0answers 52 views ### isomorphism of algebric torus I'm trying to prove the following: Let D_n=(\mathbb{C}^{\times})^n (an algebric torus of rank n). Assuming D_k is isomorphic to D_n as an algebric group. Prove that k=n. So far, I managed ... 1answer 42 views ### Compactness and sequential compactness in metric spaces I got a question: I'm trying to proof that every metric space is compact if and only if the space is sequentially compact. In all the proves I have found, they used the Bolzano-Weierstrass theorem. Is ... 1answer 55 views ### Question about Alternating forms So I understand the definition of an alternating form on \mathbb{R}^m, but I don't really understand the proof of the lemma. Could someone explain the first observation? Why is it so? 1answer 41 views ### How to construct/describe the set of unions of all intervals (a, b) of the real line? I have this problem in my textbook where I should verify whether a set \Omega is a topological structure in X=\mathbb R. \Omega is said to be "the set of unions of all intervals (a, b): a, b ... 1answer 46 views ### Prove that \overline{Int\,(\overline{Int\,A})} = \overline{Int\,A} I need to proof this statement, and I don't know where to start. In every topological sapce, we have that \overline{Int\,(\overline{Int\,A})} = \overline{Int\,A} I tried to show that ... 1answer 21 views ### Proving that b \in \overline{A} if and only if \rho(b,A) = 0 I need some help with this problem: Let be (X,\rho) a metric space, A \subseteq X and b \in X. The distance from A to b is defined as \rho(b,A) = \inf\,\{ \rho(b,a) : a \in A \}. Prove ... 0answers 54 views ### “forgetting base point”-map properties I need help with the following problem: Let P:\pi_1(X,x_0)\to S(X) be the map from the set of homotopy classes of loops based at x_0 to the set of homotopy classes without restriction on the ... 4answers 65 views ### Closure of union of two sets Show that$$\overline{A\cup B} = \overline A\cup\overline B$$\overline A=A\cup A' where A' are the limit points My attempt: Since \overline A is closed and \overline B, the union of two ... 0answers 37 views ### Classic applications of Baire category theorem [duplicate] Problem: Suppose that f:\Re^+\to\Re^+ is a continuous function with the following property: for all x\in\Re^+, the sequence f(x), f(2x),f(3x)\cdots tends to 0. Prove that ... 0answers 42 views ### Showing the bijection to prove that two sets are equivalents I need to prove that two sets X and Y are equivalent by showing the bijection between them, where X = \{x: Ax = c\} and Y = \{y: Ay\le b,-Ay\le -b\}. A is a matrix, x,y,b and c are ... 1answer 25 views ### Show that set is path connected? How do I show that the set A = \{(x,y) \in R^2: x \geq 0, y \geq 0\} \cup \{(x,y) \in R^2: x \leq 0, y \leq 0\} is path connected. I know that I need to construct a continuous function f:[0,1] ... 1answer 15 views ### Neighbourhood filter of an isolated point of a topological space How can I prove this? Let (X,\tau) a topological space, and x \in X. Then the neighbourhood filter \mathcal{V}(x) is an ultrafilter if and only if x is an isolated point of X Thanks a ... 1answer 35 views ### Find a topological space such that (A^a)^a \nsubseteq A^a I'm having some troubles with this problem, I hope you could help me: "Given (X,\tau) a topological space, find A \subseteq (X,\tau) such that (A^a)^a \nsubseteq A^a, where A^a = \{ x \in A : ... 1answer 33 views ### A Property equivalent to being a closed map Please, I like you to view this statement and tell if I'm doing something wrong. Proposition: A function f:X \rightarrow Y between topological spaces is closed if and only if for all A \subseteq ... 1answer 20 views ### Set of limit points on a topological spaces. I like to solve this problem. "Problem: In any topological space, the set of limit points of a sequence is closed." The proof is easy when we work with metric spaces, but how can I generalize this ... 1answer 42 views ### The metric space containing a compact subset is separable Let (X,d) be a metric space and K be a cpt subset of X. If it is possible to derive 'X is compact', then since compact metric space is separable, X is separable. But I'm not sure that X is compact. Do ... 1answer 54 views ### Questions on connectedness I have a final examination in general topology this week, and I've been doing past papers for the past two days in anticipation for it. I'm not sure if my answers are correct so could someone tell me ... 0answers 21 views ### Prove that if p: Y \to X is a covering space and X is path connected, then the cardinality of p^{-1}(X) is constant. [duplicate] Let p: Y \to X be a covering space and X is connected. I want to show that \forall x \in X the cardinality of p^{-x} is the same. \textbf{My Attempt:} Let us first fix a point x_0 \in X ... 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} $$Proposition: The intersection of ... 1answer 39 views ### Does there exist a subset S\subset\mathbb R such that inf \{a>0:S+a=\mathbb R-S\}=0? I founded the following question a good challenge in real analysis and topological properties of real line... Does there exist a subset S\subset\mathbb R such that inf \{a>0:S+a=\mathbb ... 1answer 23 views ### Some Topological Properties of Starlike Sets! A subset E of \mathbb R^n is starlike if it contains a point p_0 (called a center for E) such that for each q\in E, the segment between p_0 and q lies in E. For more information please ... 0answers 47 views ### Prove that if f:D^2\to D^2 is a homeomorphism, then f(S^1)=S^1 I've already proved that set of points z\in D^2 such that D^2-z is simply connected is precisely S^1. Now from this, I'm supposed to conclude that if f:D^2\to D^2 is a homeomorphism, then ... 0answers 22 views ### Find a circle which is a strong deformation retract of \mathbb{R^2}-x_0 Let x_o\in\mathbb{R^2}. Find a circle which is a strong deformation retract of \mathbb{R^2}-x_0 Proof: If x_0=(a,b), Let S=\{(x-a)^2+(y-b)^2=1\}. Then f_t(r,\theta) = (\exp((1-t)r),\theta), ... 1answer 20 views ### Prove that h and p*q are homotopic relative to {0,1} Let 0<s<1. Given paths p and q with p(1)=q(0), define h by the formula$$h(t) = \begin{cases} p(t/s),& \text{if} \quad 0 \leq t \leq s \\ q((t-s)/(1-s)), &\text{if} \quad ...
Let $p:D^3\to \mathrm{SO}(3)$, where $p(x)$ is the identity matrix if $x=0$, otherwise $p(x)$ is the rotation by the angle $\pi ||x||$ around the line connecting the origin and $x$. Assume without ...