Questions tagged [general-topology]

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

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0answers
38 views

Why is this a dense set? (Lemma 6.13, Lee ISM)

In Lemma 6.13 of Lee's 'Introduction to Smooth Manifolds' Lee is showing that the images of two functions $\kappa, \tau$ are subsets of measure zero in $\mathbb{RP}^{N-1}$. It follows then that their ...
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1answer
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Is there something missing in this exercise?

Let $n \in \mathbb{N^{*}}$ and $||\cdot||$ be a $\mathbb{R^n}$ norm. Let $F \subset \mathbb{R^n}$ be a closed set. I have to prove that $F_{\epsilon}:={\{x\in \mathbb{R^n}:d(x,F)\leq \epsilon\}}$ is ...
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1answer
23 views

Embedding/map that respects boundaries?

What do you call an embedding (or map in general) from a manifold to a manifold $i: M \rightarrow N$, such that $i(\partial M) \subseteq \partial N$? Is there a difference, if we define it as a ...
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1answer
34 views

Base in topological space

Let $B_1$ and $B_2$ be the basis in the topological space $(X,\tau)$. Will $B_1\cup B_2$ and $B_1\cap B_2$ be the base? I think intuitively it would be $B_1\cup B_2$. But $B_1\cap B_2$ may not be ...
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About Density and continuous and open function.

I've seen a proposition like that Proposition: If $f:X\longrightarrow Y$ is an open and continuous function and $D$ is dense in $Y$, then $f^{-1}(D)$ is dense in $X$. Is that proposition correct? if ...
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1answer
29 views

Prove that a set $Y$ in a metric space $(X, d)$ is open if and only if it contains none of its boundary points.

I will preface this question by saying that I realise that there are a lot of similar questions to this one, which have already been answered on this site, but none, which are exactly this question. ...
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26 views

If $L$ is a straight line then what is the topology that $L$ inherits as a subspace from $\mathbb{R}_l \times \mathbb{R}$

If $L$ is a straight line then what is the topology that $L$ inherits as a subspace from $\mathbb{R}_l \times \mathbb{R}$ and as a subspace of $\mathbb{R}_l \times \mathbb{R}_l$ . I know that the ...
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2answers
132 views

Problem $2.18$, Rudin's RCA - Painfully Set Theoretic

Problem $2.18$: This exercise requires more set-theoretic skill than the preceding ones. Let $X$ be a well-ordered uncountable set which has a last element $\omega_1$ such that every predecessor of $\...
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33 views

Let $X$ be an ordered set .If $Y$ is a proper subset of $X$ that is convex in X,then is Y an interval or ray in $X$?

The question is from Munkres and this has been answered a lot of times . However, the problems that I am facing are : How does a ray look like in $\mathbb{R}^3$ or $\mathbb{R}^2$ ? What is an ...
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1answer
27 views

Trying to understand a proof for the limit points of $\cup_{n=0}^{\infty}K_n, K_n = \{(1/n) + (1/m)\mid m=n,n+1,\dots\}$

While reviewing my answer to the exercise of 13 of chapter 2 of Baby Rudin I happened to stumble across a detailed solution manual by Kit-Wing Yu, A Complete Solution Guide to Principles of ...
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In metric space, countable union of compact sets is separable

I'm trying to prove the next statement: If $(X,d)$ is a metric space and $K_n\subseteq X$ a compact sub-set for every $n\in\mathbb{N}$ then: $\bigcup_{n=1}^{\infty}K_n$ is separable. In my attempt I ...
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1answer
34 views

Intersection of images of balls

Let $f: M \rightarrow M$ be continuous. For every intrger $n\geq0$, let $f^{n}: M \rightarrow M$ be defined such that $f^{0}=i d$ and $f^{n+1}=f \circ f^{n}$. Supose that for some $a \in M$, $m \neq n ...
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1answer
41 views

About a perfect set that is disjoint with family in the form $A\cdot\Bbb Q$ where $A$ is a finite set. [closed]

$A\subset\Bbb R$ is perfect if it is a closed set and contains no isolated point. let $A,B$ be non-empty subsets of $\Bbb R$ and by $A\cdot B$, we mean that $A\cdot B=\{a\cdot b\colon \forall a\in A \ ...
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26 views

Winding of a series of complex terms $\exp(i k_n x)$ with incommensurate frequencies $k_n$

Assumptions and definition of the problem: I consider the complex function $$ f(x) = \sum_{n=0}^M a_n \exp(-i k_n x), $$ where $M>2$ is a finite integer, $x$ is a real-valued number, $k_n$ is a set ...
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1answer
48 views

$f^{-1}(U)=S$ always open in $S$

I follow the proof for (a) but am unsure on a thing. But why is $f^{-1}(U)=S$ always open in $S$? I mean that would mean $S$ is open but why is $S$ open? credit to Hawaii.edu
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A Proof of a Theorem by Burgess

Consider the following result by Burgess: Theorem: Let $E$ be an analytic equivalence relation on a Polish space $X$. Then either $|X/E|\leq\aleph_1$ or there is a perfect set of pairwise $E$ ...
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Is $[0,1]=\cup_{i=1}^n E_i$, $\operatorname{Int}(E_i)=\emptyset$ possible?

I am wondering if it is possible to divide $[0,1]$ into finitely many subsets $E_i$, $1\leq i\leq n$, each $E_i$ has empty interior. If this is possible, then furthermore is it possible that $x=0$ is ...
2
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1answer
86 views

Locally Euclidean topology - but not Hausdorff

We consider the set $X=\mathbb{R}\cup \{\star\}$, i.e. $X$ consists of $\mathbb{R}$ and an additional point $\star$. We say that $U\subset X$ is open if: (a) For each point $x\in U\cap \mathbb{R}$ ...
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1answer
46 views

The range of the function $F$ is $S^2\setminus \{\textbf{n}\}$

Let $S^2:=\{(x,y,z)\in \mathbb{R}^3 \mid x^2+y^2+z^2=1\}$ be the unit sphere, $\textbf{n}:=(0,0,1)$ the northpole of $S^2$ and $\textbf{s}:=(0,0,-1)$ the southpole of $S^2$. Let $F:\mathbb{R}^2\...
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1answer
26 views

Is the following an open cover for the set $K = \{0\}\cup \{1/n\mid n \in \mathbb{N}\}$?

I know it has been asked to death on this site how to prove that $K = \{0\}\cup \{1/n\mid n \in \mathbb{N}\}$ is compact. I would like to not be spoiled about the proof as IMO my question is ...
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1answer
37 views

Second Countability Implies Separability

Munkres states explicitly that: Theorem 30.3: Suppose that $X$ has a countable basis. Then: (b) There exists a countable subset of $X$ that is dense in $X$. The following proof he gives is ...
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Proving semiregularity of a topological space.

I'm working in some exercise of the book Extensions and Absolutes of Hausdorff Spaces by Porter and Woods and I'm stucked in the proof. The exercise is $7$H(2): Let $X$ be a regular space, $Y$ a set ...
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57 views

Are the algebraic numbers dense in $\mathbb{C}$? [duplicate]

Since rational numbers are real algebraic numbers, clearly the real algebraic numbers must be dense in $\mathbb{R}$. So it seems natural to ask: are the (complex) algebraic numbers dense in $\mathbb{C}...
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Set B(x,y) Norm Space [closed]

Counter example for which $X$ is a Banach space and $Y$ is not a Banach space. And $B(X,Y)$ is not complete.
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18 views

How does one establish matrix similarity in a metrical fashion?

Suppose one wants a distance metric that describes the information necessary to convert one matrix into another. How would one go about doing it? The first thing that I tried was literally that, but ...
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1answer
40 views

Is the image of a proper smooth embedding always a closed set?

Suppose $M$ is a $n$-dim. smooth manifold and $f : M \to \mathbb{R}^n$ is a smooth embedding of $M$ into some Euclidean space. I have two related questions. I have read on this site that a smooth ...
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Understanding definitions of Cylinder Topology and Borel Cylinder $\sigma$-algebra.

I'm trying to understand the definition of Cylinder Topology, and Borel Cylinder $\sigma$-algebra, in the picture below, and what it's described in this wikipedia page. Let $(\mathbb{R}^d)^{\mathbb{T}...
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21 views

Canonical LF topology

Wikipedia: The space of test functions usually consists of smooth functions with compact support that are defined on some given non-empty open subset ${\displaystyle U\subseteq \mathbb {R} ^{n}}$. ...
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30 views

X is a connected and compact Hausdorff space with some properties.Prove that X is homeomorphic to $\mathbb{S}^1$.

$X$ is a connected and compact Hausdorff space,and for all $x \in X$,there is a open neighborhood of $x$ homeomorphic to $(-1,1)$.Prove that $X$ is homeomorphic to $\mathbb{S}^1$. At first,I replace X ...
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1answer
46 views

Show that if $A$ is a basis for a topology on $X$,then what will be the topology generated by $A$

Show that if $A$ is a basis for a topology on $X$,then the topology generated by $A$ equals the intersection of all the topologies on $X$ that contain $A$.Prove the same if $A$ is a sub basis. My ...
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1answer
60 views

the minimal uncountable well-ordered set $S_\Omega$ and the sequence lemma (example 3, sec 28 in Munkres topology)

First How do prove that $S_\Omega$ satisfies the sequence lemma? Munkres says 'you can readily check', but it is not easy for me. Second How does the fact that there is no sequence of $S_\Omega$ ...
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1answer
54 views

Baby Rudin Definition 2.18 (i) : Bounded set

I am trying to self study Rudin's Principles of Mathematical Analysis and I have been stuck on the definition of bounded sets stated in the book: $E$ is bounded if there is a real number $M$ and a ...
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Partial Converse of Uniform Limit Theorem from Munkres' Topology

Here is the question given in the book Topology written by James R. Munkres. Prove the following partial converse of the uniform limit theorem: Let $f_n : X \to \mathbb{R}$ be a sequence of ...
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Let $A \subseteq X$ be closed and $X$ compact with $A \neq \emptyset \neq X$. Is $C \cap \partial A \neq \emptyset$ for each component $C$ of $A$?

I know the result is true if $X$ is additionally assumed to be connected. Is the claim false without the connectedness? I can't think of a counter-example.
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Conceptual proof of Riemann–Hurwitz formula

Lately I came across the very interesting Riemann–Hurwitz formula. I believe I understand the claim of the formula, but I do not understand well why this formula is true. I am looking for some ...
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1answer
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What sorts of (sets of) equations are “approximately compatible” with the $2$-sphere?

Given a metric space $\mathcal{X}=(X,d)$ and an equational theory (in the sense of universal algebra) $\mathsf{E}$, say that an approximate model of $\mathsf{E}$ on $\mathcal{X}$ is a sequence $(\...
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Simple Infectionmodell, recursive Sets,

i am working on this Problem: let $0 < \alpha, \gamma < 1$ moreover let $F$ be : $$F: \mathbb R^3 \rightarrow \mathbb R^3, x= \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix} \rightarrow \begin{pmatrix}...
4
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60 views

If$ \{\tau_{\alpha}\} $be a family of topology on $X$ then the smallest topology containing all $\{\tau_{\alpha}\}$

If $\{\tau_{\alpha}\}$ be a family of topology on $X$ then the smallest topology containing all $\{\tau_{\alpha}\}$ The basic intuition that I have is the smallest topology will be the topology ...
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1answer
22 views

Removing the upper and the lower part of the sphere gives a cylinder

Consider the sphere $S^n$ and let $0 < \epsilon < 1$. We write $$U_\epsilon:= \{x \in S_n: x_{n+1} \le \epsilon\}, \quad V_\epsilon: = \{x \in S^n: x_{n+1}\ge -\epsilon\}$$ Is it true that $U_\...
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3answers
88 views

How does rotating a circle create a spherical surface without leaving “gaps”?

EDIT: Agreed, this isn't a well formed question. But responses below have at least given me a different way to think about it. First post here because I'm not a math guy, but I have a feeling the ...
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0answers
41 views

extension to a continuous function

I self-studying Aluffi chapter 0. I need the following result to prove a exercise. Assume that K is compact topological space. Fix a $p \in K$. Assume that $g : K \rightarrow \mathbb{R}$ is continuous ...
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1answer
30 views

On the basis for quotient topology

There is some discussion on this topic already in Basis for the Quotient Topology. But I had some other questions. If I start with a basis $\mathcal{B}$ in the original topological space X and ...
3
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2answers
79 views

Homeomorphism between two disks with a hole

I need to show that $\mathbb{D}^2\setminus B(x_1,r_1)$ and $\mathbb{D}^2\setminus B(x_2,r_2)$ are homeomorphic. I have tried with a Möbius transformation of the disk but I can't show that it sends the ...
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1answer
66 views

why $U_{y_1}\cap …\cap U_{y_n}$? why not $U_{y_1}\cup …\cup U_{y_n}?$

I have some confusion about the statement in Munkras Book Theorem $26.3$: Every compact subspace of a hausdorff space is closed In the theorem of the proof it is written that the open set $V_{y_1} ...
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1answer
20 views

Checking connectedness by an associated graph

Let $X$ be a topological space with an open cover $\{U_\alpha\}$ such that each $U_\alpha$ is connected and nonempty. Form a graph as follows: for each $\alpha$ put a vertex $v_\alpha$. Also $v_\alpha$...
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1answer
46 views

Consider nine topologies on $X=\{a,b,c\}$ then which of them are comparable?

I am very new to topology.The points which are not clear to me is that: 1)There are $29$ possible ways in which sets can be considered then how do we conclude which of them are topology and which of ...
2
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2answers
58 views

What does this property of the basis try to generalize?

Given a topological space $X$ and $x \subset B_1 \cap B_2$ where $B_1$ and $B_2$ are basis elements then there exists a basis $B_3$ such that $ x \in B_3 \subset B_1 \cap B_2$.Is it trying to ...
0
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1answer
31 views

Bijection between compact space 𝐾 and maximal ideals of real-valued functions on K

I know this question has been asked before, because as I was gonna ask it. A similar question popped up. I just have different question regarding the details of the proof. I am currently studying ...
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0answers
50 views

Munkres, second half of Lemma 13.2

Lemma 13.2 in Munkres says: Let $X$ be a topological space. Suppose that $\mathcal{C}$ is a collection of open sets of $X$ such that for each open set $U$ of $X$ and each $x \in U$, there is an ...

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