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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0
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2answers
101 views

Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives.

Let U be an open subset of $\mathbb{R}^n$ and C a compact subset of U. Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives. Prove that f is Lipschitz on C. Thoughts: Let ...
4
votes
0answers
36 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 ...
1
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4answers
90 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 ...
3
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3answers
1k views

Is product of two closed sets closed?

If $A$ is closed in $X$ and $B$ is closed in $Y$, does it follow that $A \times B$ is closed in $X \times Y$ ?
4
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0answers
30 views

Spaces whose all their metrizations are complete [duplicate]

Which metrizable topological spaces $(X,\tau)$ posses the following property: Every compatible metric (i.e one which induces the same topology $\tau$) is complete. Compact metrizable spaces satisfy ...
0
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3answers
49 views

$\mathbb{N}$- a complete metric space with $d(x,y)=|x-y|$

$\mathbb{N}$- a complete metric space with $d(x,y)=|x-y|.$ This seems quite intuitively correct, but I do not know how to prove this formally, does anyone know how they would go about this?
1
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2answers
31 views

Show $f: S^1 - {N} \to \mathbb{R} $ $f(x_1,x_2) = \frac{x_1}{1-x_2}$ is Homeomorphism

$S^1$ is a unit circle and $N := \{ (0,1) \in S^1\}$. The question hints that the for any $(x_1,x_2) \in S^1- {N}$, line joining $N$ and $( x_ 1 , x_ 2 )$ meets the $x$ -axis at ($f ( x_ 1 ;x_ 2 ) , 0 ...
3
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1answer
22 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 ...
1
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0answers
20 views

commutativity taking the complement and taking fibers

Let $\mathcal M \rightarrow S$ be 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 ...
4
votes
1answer
60 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. ...
3
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3answers
97 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 $S^2$. ...
0
votes
1answer
40 views

Any filter is a neighborhood filter with a fixed point removed from all neighborhoods. Is the topology which produces this neighborhood unique?

Assume $\mathcal F$ is a filter on the set $X$ and $\infty \notin X$ and $$Y=X\cup \{\infty\}$$ We know there's some topology $\mathcal T$ on $Y$ such that $$\mathcal F =\{U\cap X \mid U \text{ is a ...
3
votes
0answers
48 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
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0answers
29 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 ...
0
votes
2answers
91 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 ...
6
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4answers
92 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 ...
2
votes
1answer
66 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 ...
1
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2answers
84 views

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 ...
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 ...
4
votes
1answer
62 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 ...
1
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0answers
26 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 $X$ ...
5
votes
2answers
128 views

Is $\mathcal P(X)$ connected when $(X,\mathcal P(X),m)$ is a measure space and $P(X)$ is equipped with the metric $d(A,B) =m(A\Delta B)$?

Is $\mathcal P(X)$ connected when $(X,\mathcal P(X),m)$ is a measure space and $P(X)$ is equipped with the metric $d(A,B) =m(A\Delta B)$? Think when we look at the equivalence classes of almost ...
2
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0answers
40 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, ...
11
votes
5answers
582 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 ...
13
votes
3answers
369 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 ...
13
votes
2answers
249 views

Are compact spaces characterized by “closed maps to Hausdorff spaces”?

It is well known that any continuous map from a compact space to a Hausdorff space must be a closed map. Does this fact characterize compactness? That is, if for a space $X$, every continuous map to ...
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)|+ ...
2
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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 ...
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 ...
8
votes
1answer
487 views

Homeomorphisms of X form a topological group

So I'm just learning about the compact-open topology and am trying to show that for a compact, Hausdorff space ,$X$, the group of homeomorphisms of $X$, $H(X)$, is a topological group with the compact ...
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 ...
4
votes
1answer
30 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 ...
4
votes
0answers
37 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 ...
1
vote
1answer
29 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, ...
12
votes
2answers
2k views
+100

Normal subgroups of free groups: finitely generated $\implies$ finite index.

I am looking at what should be a simple exercise in geometric group theory. I have reduced the problem to just completing an exercise from Hatcher, Section 1.B page 87: 7. If $F$ is a finitely ...
7
votes
1answer
235 views

“Wheel Theory”, Extended Reals, Limits, and “Nullity”: Can DNE limits be made to equal the element “$0/0$”?

"Wheels" are a little-known kind of algebraic structure: They modify the concept of a field or a ring in such a way that division by any element is possible, including division by zero, while also ...
2
votes
1answer
30 views

Circloid with non-empty interior

By a "circloid" I mean a continuum in the plane which separates the plane into exactly two components and minimal with respect to these properties, i.e. has no proper subcontinuum which separates the ...
8
votes
1answer
236 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
2answers
144 views

Lang's treatment of product of Radon measures

Let $X$ be a locally compact Hausdorff space. We denote by $\mathcal B(X)$ the $\sigma$-algebra of Borel sets of $X$. A positive Radon measure $\mu$ on $X$ is a measure defined on $\mathcal B(X)$ with ...
2
votes
2answers
78 views

Prove that a finite union of closed sets is also closed (using limit points)

Let $F_i$ be a family of closed sets, then we know that $\bigcup_{i=1}^nF_i$ is closed. Proving that statement is equivalent to proving: If $p$ is a limit point of $\bigcup_{i=1}^nF_i$ then ...
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 ...
0
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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 ...
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
24 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
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 ...
3
votes
2answers
2k views

Infimum is a continuous function, compact set

Let $f: X \times Y \rightarrow \mathbb{R}$ be a continuous map. Show that if $Y$ is compact then the function $g: X \rightarrow \mathbb{R}$ defined by $g(x) = \inf \{f(x,y): y \in Y\}$ is also ...
1
vote
2answers
42 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 ...
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 ...