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; ...

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3
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3answers
87 views

Show that $\mathbb{R}^2/{\sim}$ is homeomorphic to the sphere $S^2$.

$\mathbb{R}^2/{\sim}$ is the smallest equivalence relation such that $P\sim Q$ for all $P,Q$ with $\|P\|_2,\|Q\|_2 \geq 1$. Show that $\mathbb{R}^2/{\sim}$ is homeomorphic to the sphere $S^2$. ...
1
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0answers
13 views

commutativity taking the complement and taking fibers

Let $\mathcal M \rightarrow S$ a projective irreducible scheme over the spectrum of a dvr and $U\subset \mathcal M$ an open subscheme surjective on $S$. Is it true for both points (generic and closed) ...
3
votes
0answers
42 views

How to draw the knot 2, -32, 41?

Hej, I have the following exercise: Draw the tangle 2, -32, 41 and the corresponding knots obtained by connecting the NW string to the NE string and the SW string to the SE string. (From C. C. ...
1
vote
0answers
25 views

How do I prove that this product space is normal?

Let $A$ be a compact subspace of $\mathbb{R}^2\setminus\{0\}$. Let $C$ be a connected component of $\mathbb{R}^2\setminus A$. Define $D=C\cup A$. How do I prove that $D\times [0,1]$ is normal? I ...
1
vote
0answers
8 views

Are Homogenous countable complete metric spaces always discrete?

Let $M$ be a countable complete metric space such that the group of isometries of $M$, $Iso(M)$ acts transitively on the points in $M$. Does it follow that the topology induced by the metric is the ...
0
votes
2answers
88 views

Why is every point in an open interval $(a,b)$ not a limit point?

If I have an open interval $\Bbb R\supset A=(a,b)$ then I can pick any $x:a<x<b$ and make a ball with center $x$ which contains a point inside the interval. However, this article from proofwiki ...
2
votes
1answer
64 views

Dedekind Construction Of Real Numbers

If we define Dedekind-real numbers as Dedekind cuts, i.e. $\sqrt 2 = \{\text{rationals less than }\sqrt2\} \cup \{\text{rationals more than } \sqrt2\}$, can we define addition and multiplication of ...
4
votes
1answer
56 views

Can we parameterize a topological space?

It has been few months since I started doing topology . There was this idea which struck me a few days ago . For example the parametrization of a line is $$x=qv+a,$$ where $t$ is the parameter. ...
1
vote
0answers
22 views

Is $\mathbb R^N$ an $C$-distinguished topological space?

I am reading a paper which has some complicated construction on a Hausdorff topological space called $C$-distinguished topological space. The paper says that a $C$-distinguished topological space $X$ ...
6
votes
4answers
88 views

Definition of Equivalent Norms

Two norms $F,G$ are equivalent when there are constants $a,b$ such that $aF \le G \le bF$. I'm reading about this idea, and so far I've seen that equivalence of norms implies that the underlying ...
3
votes
1answer
56 views

Is Heisenberg group Euclidean?

I'm reading an article speaking about Heisenberg group $\mathbb H^n$ and some of its properties. Now, I have some questions to ask, hoping to be clear enought. Reading the introduction I've ...
2
votes
0answers
36 views

Topologies on the collection of $\sigma$-algebras

Let $X$ be a non-empty set and let $\mathfrak S$ be the collection of all $\sigma$-algebras on $X$. That is, a typical element $\mathscr S\in\mathfrak S$ is a $\sigma$-algebra on $X$. For example, ...
4
votes
0answers
23 views

How are the pseudo-Riemannian metric tensor properties restricted by the manifold topology in pseudo-Riemannian manifolds?

My understanding is that a pseudo-Riemannian metric tensor induces a topology that is not compatible with the manifold topology, and obviously the manifold topology prevails if we are to have a ...
13
votes
3answers
366 views

Is a space compact iff it is closed as a subspace of any other space?

I am trying to come up with an alternate definition of a compact topological space that coincides with the usual one. Sorry if my topology is a little rusty. My proposed alternative definition is ...
2
votes
1answer
34 views

Specific question on $l^p$ spaces and its dual in weak * topology

I am covering now Lp spaces in my summer real analysis course and this problem from Folland related to the dual of Lp stumped me hard, it is problem 19 chapter 6 reads as follows: We define $ ...
3
votes
0answers
40 views

Proving that $S^c=\left\{f(x)\in C^2[0,1]\;\Big\vert\; \int_{0}^{1}f(x)dx > 3\right\}$ is open in $C^2[0,1]$ with a specific metric

I am trying to prove that $$S^c=\left\{f(x)\in C^2[0,1]\;\Big\vert\; \int_{0}^{1}f(x)dx > 3\right\}$$ is open in $C^2[0,1]$ with the metric $d$ given by $$ d(f,g)=\sup_{x \in [0,1]}|f(x)-g(x)|+ ...
0
votes
1answer
27 views

Feedback on my solution “Determine a set compact or not”

Let $X :=\{(x_1,0,x_2) \in \mathbb{R^3}, x_1, x_2 \in \mathbb{R}\}$ $\mathcal{T}$ be subsapce topology coming from standard topology on $\mathbb{R^3}$. My answer is that it's compact. Reason: Define ...
2
votes
1answer
39 views

Completeness of ${C^2[0,1]}$ with under a specific metric

Prove that ${C^2[0,1]} $ (set of two times differentiable functions)is complete with metric: $$d(f,g)=\sup_{x \in [0,1]}|f(x)-g(x)|+ \sup_{x \in [0,1]}|f'(x)-g'(x)| + \sup_{x \in ...
10
votes
5answers
579 views

Why formulate continuity in terms of pre-images instead of image?

I wanted to discuss my intuition of why we formulate the concept of continuity in terms of pre-image of open set is open instead of images for example if we consider $f(x) = c$ where $c$ is some ...
4
votes
1answer
29 views

Topologically distinguishing Mobius Strips based on the number of half-twists

We can distinguish between a (closed) Mobius strip and 'regular' (untwisted) strip by examining the set of points which have no neighborhood homeomorphic to a disk (intuitively, the 'boundary' of the ...
3
votes
0answers
34 views

What are the universally effective epimorphisms of topological spaces?

An effective epimorphism in a category is a morphism that is the coequaliser of its kernel pair, and a universally effective epimorphism is a morphism $f : X \to Y$ such that, for every pullback ...
4
votes
1answer
61 views

Functorial properties of the compact open topology.

Let $X,Y,Y'$ be topological spaces and $A\subseteq Y$ a subspace. Every set of continuous maps is equipped with the compact-open topology. Is the canonical map ...
2
votes
1answer
42 views

Questions about the Identification Topology and Equivalence Class from “Introduction to Topology” by Mendelson

I am currently reading Introduction to Topology by Bert Mendelson, and I have some questions regarding the topic on Identification Topology in his book. Let $(X,\tau)$ and $(Y,\gamma)$ be topological ...
8
votes
1answer
235 views

What star domain has a non-star-domain interior?

Definition: We call a subset $S$ of $\mathbb{R}^n$ a star domain (or star-shaped) if there exists a point $x_0 \in S$ such that for every $x \in S$, the line segment $\overline{x_0x}$ is contained ...
0
votes
1answer
26 views

Dense Domain: Preimage

Given Banach spaces $X$ and $Y$. Regard a bounded operator: $$A\in\mathcal{B}(X,Y)\implies A\in\mathcal{C}(X,Y)$$ Then for dense sets: $$W\leq Y:\quad \overline{W}=Y\implies\overline{A^{-1}W}=X$$ ...
1
vote
1answer
28 views

How do I show that this topology on this linearly-ordered set is regular?

Given some linear ordered set $X$, we define a topology by the basis: all sets of the form $(a,b)$ or $(a,\infty)$ or $(-\infty,b)$, where $a,b \in X$. I need to prove that this topology is regular, ...
0
votes
0answers
28 views

Plotting Distance Constrained Points on a Plane

Does anybody know of some algorithmic way to tell if it is possible to plot a set of distance constrained points on a cartesian plane. Or, better still, a method to determine the minimum number of ...
1
vote
1answer
30 views

If $A\subseteq\Bbb R$ is nonempty with $|A|\ge 2$, then $A$ totally disconnected $\iff A^\circ=\emptyset$.

In the course of working on an exercise, I came up with the claim given in the title. Just looking for verification. $\underline{\text{Claim: } A\text{ is totally disconnected}\iff ...
1
vote
2answers
82 views
+100

Euclidean Spaces: Embedding

Given the real line $\mathbb{R}$ and plane $\mathbb{R}^2$. Are there maps: $$\eta\in\mathcal{C}(\mathbb{R}^2,\mathbb{R}),\vartheta\in\mathcal{C}(\mathbb{R},\mathbb{R}^2):\quad ...
0
votes
1answer
34 views

Problem in showing that a sequence is a Cauchy sequence on a space with the integral metric.

I'm having difficulty following what is going on and understanding parts in the following example. It is quite similar to an example I posted before (Changing of the limits of integration with the ...
1
vote
2answers
23 views

denseness of polynomials in bounded borel measurable functions

Let $K\subseteq \mathbb{R}$ be compact, consider $B(K)$ the set of all bounded borel measurable functions $f:K\to \mathbb{C}$ and endow $B(K)$ with the uniform norm, so you obtain a Banach space. My ...
1
vote
2answers
41 views

Why is the topology on $\mathbb{R}$ formed by the basis $[a,b)$ normal?

Why is the topology on $\mathbb{R}$ formed by the basis $[a,b)$ normal? I need to prove that two disjoint closed sets are contained wtihin two open disjoint sets. First, I tried to understand how a ...
1
vote
0answers
51 views

Cylinder and Möbius strip as fiber bundles: trivializations and cocycles

I know that this question has already been asked, but I couldn't find a clear answer. I have to show that the cylinder and the Möbius strip are fiber bundles over $S^1$ with fiber an open interval ...
2
votes
2answers
50 views

If $A$ is path connected, then $\bar A$ is path connected?

I know the topologist's sine curve serves as a counter example. But how do I show that $A = \{(x, \sin (1/x)): 0<x\le 1\}$ is path connected?
5
votes
1answer
54 views

Approximating nice functions with wild ones

Let $X$ and $Y$ be toplogical spaces, and call a function $f:X\to Y$ wild if the preimage $f^{-1}(\{y\})$ is dense in $X$ for every $y\in Y$ -- or, equivalently, if the image of every nonempty open ...
8
votes
2answers
83 views

Is the following a characterization of $\Bbb Q\cap\cal C$, where $\cal C$ is the Cantor set?

Let $A$ be an ordered set, with the following properties: $A$ is countable $A$ has a least and greatest element Between any two points with successors are points without successors; between any two ...
1
vote
4answers
82 views

Are there an infinite number of open balls in an open set in a metric space?

Let's start off by recalling the definition of an open set in a metric space: A set $A$ in a metric space $(X,d)$ is open if for each point $x\in A$ there is a number $r\gt0$ such that $B_r(x)\subset ...
1
vote
1answer
33 views

The convergence of different metrics on the same space

The following example is from my notes, and I would like clarification on some wider points connected to it, namely about extensions from what we understand metrics and metric spaces to be. It follows ...
0
votes
1answer
39 views

Topology on compactly supported smooth functions

I'm confused by a set of lecture notes I'm reading and would like help in understanding what's going on. First, there is the following nice theorem. Theorem. The topology of a locally convex space is ...
6
votes
1answer
75 views

Is there an infinite topological meadow with non-trivial topology?

For reference meadows are a generalization of fields that were designed to be compatible with the requirements of universal algebra. Specifically a meadow is a commutative ring equiped with an ...
2
votes
1answer
31 views

A covering space of CW complex has an induced CW complex structure.

Let $X$ be a $CW$ complex, and let $q : E \rightarrow X$ be a covering map. Prove that $E$ has a $CW$ decomposition for which each cell is mapped homeomorphically by $q$ onto a cell of $X$. Hint: ...
0
votes
1answer
29 views

Product-topology of discrete $\{ 0, 1 \}$ spaces

I was thinking about the following: Take the product $\prod_{i \in I} \{ 0, 1 \}$ for each $\{ 0, 1 \}$ being discrete. Is the product-topology also the discrete topology? I'd say intuitively "no", ...
0
votes
1answer
25 views

Please check if my reasoning about whether this topological space is connected is correct

$X = \mathbb{R}$, $\mathcal{T} =$ the collection of all subset $U$ of $\mathbb{R}$ such that $U = \emptyset$ or $\mathbb{R} - U$ is finite. Then, $(X,\mathcal{T})$ is connected. My thoughts: assume ...
1
vote
1answer
122 views

Cauchy Sequences--is the floor function of a Cauchy sequence also a Cauchy sequence?

Okay so say you have some Cauchy sequence (a_n). And c_n=[[a_n]], where [[x]] refers to the greatest integer less than or equal to x. Is c_n also a Cauchy sequence? This is what I've got so far, ...
7
votes
2answers
115 views

Proving that the product of two numbers (in $\mathbb{R}$ or $\mathbb{C}$) is a continuous function.

This is what is given in the textbook, I will highlight what is confusing me: Product in field $\mathbb R$ or $\mathbb C$,on $X \times X$ defined as: $$(x,y)\mapsto xy$$ (Let indicate that map with ...
-2
votes
0answers
61 views

How to prove that multiplication is continuos function? [closed]

How to prove that multiplication is continuous function? $x \rightarrow x^n$ Can somebody help me? :)
4
votes
1answer
34 views

$\mathbb{F}_p[T, 1/T]$ is discrete in $\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T))$, adeles.

Let $p$ be a prime number. How do I show that $\mathbb{F}_p[T, 1/T]$ is discrete in $\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T))$?
1
vote
2answers
60 views

Is the Dikin Ellipsoid actually a ball?

I have the inequality (row wise): $Ax \leq b$ The Dikin ellipsoid centered at $x_0$ with radius $r$ is: $$\{z \quad | \quad (z-x_0)^T(z-x_0) \leq \frac{r^2}{H(x_0)}\}$$ where, $$H(x_0) = \sum ...
3
votes
1answer
36 views

The same topologies

Let $L^1 (\mathbb{Z})$ be the space of all functions $f:\mathbb{Z}\rightarrow \mathbb{C}$ such that $\left\{\|f\|=\sum_{k\in \mathbb{Z}}|f(k)|<\infty\right\}$. Clearly, $L^1 (\mathbb{Z})$ is a ...
0
votes
1answer
34 views

Stein & Shakarchi, Complex Analysis, Ch.3 Ex.7

Suppose $f : \mathbb{D} \to \mathbb{C}$ is holomorphic, and $d = \sup_{z,w \in \mathbb{D}} |f(z) - f(w)|$. Show that $$ 2 |f'(0)| \leq d$$ This entire exercise is a complete mystery to me and I am ...