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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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10k views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
114
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0answers
1k views

What functions can be made continuous by “mixing up their domain”?

Definition. A function $f:\Bbb R\to\Bbb R$ will be called potentially continuous if there is a bijection $\phi:\Bbb R\to\Bbb R$ so that $f\circ \phi$ is continuous. So one could say a potentially ...
19
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0answers
429 views

How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
18
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0answers
469 views

defining a topology by its compact sets

The goal. Let $X$ be a set endowed with Hausdorff topologies $\tau_w$ and $\tau_n$, such that $\tau_w\subseteq\tau_n$. Let $\mathscr{C}$ denote a family of subsets $A\subseteq X$, which satisfies ...
17
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0answers
346 views

Is the set $\{X \in \mathcal{M}({m \times n}) : \rho(M-NX) < 1\} $ connected?

Suppose $M \in \mathcal M(n \times n; \mathbb R)$ and $N \in \mathcal M(n \times m; \mathbb R)$ are fixed with $N\neq 0$. Let \begin{align*} E = \{X \in \mathcal{M}(m \times n; \mathbb R) : \rho(M-...
16
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0answers
498 views

Exercise in Engelking's book regarding a disconnected space.

This is related to 6.3.24 in Engelking's Topology book. The Hilbert space $H$ is the set of sequences $(x_i) \in \mathbb R ^\omega$ such that $\|x\|=\sum _{i=1} ^\infty x_i ^2<\infty$, with ...
14
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0answers
254 views

What is the structure preserved by strong equivalence of metrics?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are (topologically) equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ ...
14
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0answers
410 views

Is the closure $\overline{ \{X \in \mathbb{R}^{m \times n} : \rho(M-NX) < 1\} }$ equal to $ \{X \in \mathbb{R}^{m \times n} : \rho(M-NX) \le 1\}$

Suppose $M \in \mathcal M(n \times n; \mathbb R)$ and $N \in \mathcal M(n \times m; \mathbb R)$ are fixed with $N \neq 0$. Let \begin{align*} E = \{X \in \mathcal{M}(m \times n; \mathbb R) : \rho(M-...
14
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0answers
4k views

Regarding metrizability of weak/weak* topology and separability of Banach spaces.

Let $X$ be a Banach space, $X^*$ its dual, $\mathcal{B}$ the unit ball of $X$ and $\mathcal{B}^*$ the unit ball of $X^*$. The following result is well-known Theorem. If $X$ is separable, then $\...
13
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0answers
211 views

A product topology where we allow countably many open sets

Let $\{X_{\alpha}\}$ be an uncountable collection of topological spaces indexed by the set $J$. For the space $\prod_{\alpha}X_{\alpha}$, consider the topology $\tau$ generated by the basis $$\...
13
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0answers
255 views

Norms and pointwise convergence

It is known that There is no norm $\|.\|$ on the space $E$ of continuous real-valued functions on an interval, say $[0,1]$ such that $f_n \to f$ for $\|.\|$ if and only if $f_n$ converges pointwise ...
13
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0answers
456 views

A few standard results (on metrizability and relative separation strength) without choice?

I've been going back over some results from Munkres's Topology, and I'm curious about some things.... I know that Choice principles have some connection to the separation axioms (in ZF, at least)--...
13
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0answers
188 views

Topological Space in which every compact subset is metrizable

Is there an (more or less) established name for the class of topological spaces in which every compact subset is metrizable? This is true for example in (LF)-spaces (inductive limits of Frechet-spaces)...
12
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0answers
159 views

Does the box topology have a universal property?

Given a set of topological spaces $\{X_\alpha\}$, there are two main topologies we can give to the Cartesian product $\Pi_\alpha X_\alpha$: the product topology and the box topology. The product ...
12
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0answers
204 views

Generalization of real induction for topological spaces?

Real induction is a useful proof technique which can be thought as a version of "continuous" induction. I will include here version from Pete L. Clark's text mentioned in this answer,1 where it is ...
11
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0answers
230 views

Does there exist a continuous path between two sets of oriented basis for a vector space out of a collection of subspaces?

Let $V_1, V_2, \dots, V_n$ be a collection of vector subspaces in $\mathbb R^n$. For each $j=1, \dots, n$, $\dim(V_j) = m$ with $2 \le m < n$. We also have the condition: for any collection of $\...
10
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0answers
76 views

When is uniform space normal

We know that metric spaces are normal. We also know that a uniform space is Hausdorff if the intersection of all entourages is the diagonal, in which case it is even regular. However, is there a ...
10
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0answers
357 views

Fréchet L-Spaces

NOTE: The question has now been posted on MathOverflow: Fréchet L-Spaces According to the paper The emergence of open sets, closed sets, and limit points in analysis and topology famous ...
10
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0answers
289 views

Order-preserving map of regressive functions on $\omega_1$

This questions has now been published in a journal, see update at the bottom. I posted the following question in March 2014 on MO. It did receive some attention, but the answer there remains ...
10
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0answers
845 views

Questions related to intersections of open sets and Baire spaces

EDIT: I have reposted this question on MathOverflow. (The version posted there is more concise, with some details omitted. I have also added a question about pseudobases with similar property.) MO ...
10
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0answers
2k views

Restrictions of null/meager ideal

Let I denote the null/meager ideal on reals. Is it consistent that for any pair of non null/meager sets A and B, there is a null/meager preserving bijection between A and B? In particular, is this ...
9
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77 views

If $H_1 \subset H_2 \subset G$ and $G/H_2,\ H_2/H_1$ are compact then $G/H_1$ is compact.

I'm trying to solve the following exercise of the book "Grupos de Lie - Luiz A. B. San Martin (exercise 18, page 55)": Exercise: Let $G$ be a topological group and $H_1 \subset H_2\subset G$ ...
9
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0answers
104 views

What is $\ \overline{\bigcup_{p≥ 1}\ \{A\in M_n(\mathbb C), \ A^p = I_n\}} \ $?

Let $\Gamma_p = \{A\in M_n(\mathbb C), A^p = I_n\}$ and let $\Gamma = \bigcup_{p≥ 1}\ \Gamma_p$. What is the closure of $\Gamma$ ? (This is from an oral exam). Let $B \in M_n(\mathbb C)$ such that ...
9
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0answers
175 views

Covering a set $A \subset \Bbb R$ by two families of disjoint intervals taken from given intervals

Let $A \subset \Bbb R$ be a bounded set. Every element $a \in A$ is the center of some given open interval, let's denote it by $I_a=(a-r_a, a+r_a)$. I'm interested in knowing the following: Can we ...
9
votes
0answers
441 views

Properties of King's Dream fractal

My question is focused on the King's Dream fractal, which can be defined as follows (nice pictures can be found here) : $$ \Omega = \{f^n(0.1,0.1) \;\vert\; n \in \mathbb N \} \quad f(x,y)=(\sin(...
9
votes
0answers
288 views

Simpler version of dogbone space construction

In "The cartesian product of a certain nonmanifold and a line is $E^4$" (R.H. Bing, Annals of Mathematics series 2 vol 70 1959 pp. 399–412) Bing constructs a nonmanifold, $B$, such that $B\times \Bbb ...
9
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0answers
262 views

Why use the Lefschetz Zeta function?

Given a compact, triangulable space $X$ and a continuous function $f: X\rightarrow X$, then we define the Lefschetz number $\Lambda_{f}$ by $$\Lambda_{f} = \sum_{k\geq0}(-1)^{k}Tr(f_{\ast}\vert H_{k}(...
9
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0answers
180 views

Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
9
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0answers
189 views

How to find a homeomorphism $\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ having certain properties

Let $n\ge 2$ and let $C$ be a Cantor space in $\mathbb{R}^{n}$. That is, $C$ is homeomorphic to the Cantor ternary set. Let $x$ and $y$ be two points in $\mathbb{R}^{n}-C$, and let $L_{xy}$ be the ...
8
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0answers
237 views

When does $S \cap GL_4(\mathbb R)$ have precisely two connected components where $S \subset M_4(\mathbb R)$ is a linear subspace?

This is related to the question How many connected components for the intersection $S \cap GL_n(\mathbb R)$ where $S \subset M_n(\mathbb R)$ is a linear subspace? I asked. There is a nice example in ...
8
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0answers
251 views

Given a family of 2D curves, find a 3D manifold whose geodesics project to the plane curves

Below is an image of the family of 2D curves for reference. The image is arbitrary. Still trying to formulate a concise question. I know that the geodesics for flat Euclidean Space are straight lines....
8
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0answers
234 views

When is an Open Set Homeomorphic to the Interior of its Closure?

Let $X$ be a topological space and $U \subseteq X$ open. Then $U \subseteq \operatorname{int}(\operatorname{cl}(U))$. I am looking for known assumptions on $X$ and $U$ such that one of the following ...
8
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0answers
287 views

Non empty set with zero diameter

Let $A \subset X$ where $X$ is a metric space. by definition diam$(A) = \sup\{ d(x,y), x,y \in A\}$. if $A$ is non empty and has zero diameter, can I conclude that $A$ is a singleton? i reason as ...
8
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0answers
200 views

An open and connected subset $U\subseteq \mathbb C$ is still connected if you remove a curve that lies entirely in $U$

Let $U\subseteq \mathbb C$ be open and connected. If $f\colon [0,1]\to U$ is continuous with $f(0)\neq f(1)$ and $f(s)\neq f(t)$ for $s\neq t$, then $U\setminus f([0,1])$ is connected. This seems to ...
8
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0answers
670 views

Good pair vs. cofibration

It can be shown that $i:A\hookrightarrow X$ is a closed cofibration if and only if there is a map $\varphi:X\to I=[0,1]$ and a homotopy $H:U\times I\to X$ on some neighborhood $U$ of $A$ such that $A=...
8
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0answers
205 views

Is there such a thing as 'overtification' (dual to compactification)?

The dual to the notion of compactness for spaces is overtness. This duality is not manifest in the category of spaces but rather in the quantifiers used to define these notions. Is there a process ...
8
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0answers
174 views

Flabby sheaf comonad

If one Googles sufficiently hard one finds the statement that Roger Godement gives the first example of a comonad, used to compute flabby resolutions of sheaves, in his monograph "Topologie algébrique ...
8
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0answers
2k views

Euler Characteristic of surface with boundary puncture.

I am studying a course on differential geometry. I saw the formula for the euler characteristic of a surface with $g$ holes and $b$ boundaries components and $n$ punctures is $2-2g -b +n$. In ...
7
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0answers
78 views

Let $X$ be a separable Banach space. Let $Y$ be the space formed by adding a point at infinity. Is $Y$ homeomorphic to the unit sphere of $X$?

More specifically, let $Y= X \cup \{\infty\}$ and declare open sets to be the usual open sets in $X$ together with those that are of the form $(X-C)\cup \{\infty\}$, where $C$ is norm closed and norm ...
7
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0answers
79 views

smooth or differential version of Gelfand-Neimark theorem

Using the Gelfand-Naimark theorem we can define an equivalence between the category of compact Hausdorff spaces and the category of commutative C*-algebras with unity (commC*-alg1). Someone knows if ...
7
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0answers
118 views

Are there any other fields other than $\mathbb{R},\mathbb{C}$, rich enough to have analysis built on them?

I've been thinking about this, I don't know how to look up anything similar, so here I am asking a question. Specifically, is there any space $X$ with the following properties: Algebraic structure: ...
7
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0answers
130 views

Homotopy of homeomorphisms implies homeomorphism $X \times [0,1] \to X \times [0,1]$?

Let $h_0$ and $h_1$ be self-homeomorphisms of a topological space $X$. Let us say that $h_0$ is homotopic to $h_1$, and write $h_0 \sim h_1$ if there exists a 1-parameter family $h_t$, $t \in [0,1]$ ...
7
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0answers
78 views

Are absolutely convergent series “many” or “few” compared to conditionally convergent series?

We can identify absolutely convergent series with the $l^1$ space and conditionally convergent series with a subspace of $c_0 = \{\{a_n\} \in l^\infty : a_n \to 0\}$ Since $l^1$ contains all finite ...
7
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0answers
476 views

Homeomorphism Between Quotient Space and $\Bbb{R}$

Define an equivalence relation on the plane $X =\mathbb{R^2}$ as follows: $$(x_0, y_0) \sim (x_1,y_1) \mbox{ if } x_0 + y_0^2 = x_1 +y_1^2$$ Let $X^*$ be the corresponding quotient space. ...
7
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0answers
248 views

If locally convex topologies exhibit the same dual spaces, do they exhibit the same continuous linear operators?

Consider the following setting: Let $X, Y$ be vector spaces over the field $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$. Furthermore, let $\tau_1, \tau_2$ be locally convex topologies on $X$ and $\...
7
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0answers
317 views

Density of Banach spaces

I am trying to understand the notion of density in the context of Banach spaces. The density of a topological space is the least cardinality of a dense subset. Thus, a separable Banach space has ...
7
votes
0answers
226 views

Curvature and topology

I am studying Riemannian Geometry and I came across various Theorems which give conditions on the topology of a manifold given conditions on curvature, and vice-versa. Just to mention a few of them: ...
7
votes
0answers
155 views

Examples of calculating perverse sheaves on algebraic varieties with easy stratification.

This question is also asked in mathoverflow https://mathoverflow.net/questions/232589/examples-of-calculating-perverse-sheaves-on-algebraic-varieties-with-easy-strati I have been learning ...
7
votes
0answers
459 views

Comparison of different topologies on the ordered square $[0,1]\times[0,1]$

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\T}{\mathcal{T}}$ This is textbook problem in Munkres' Topology. I know there are plenty of solutions online, but none of them are exactly the same as mine. ...
7
votes
0answers
889 views

The Weak topology on an infinite-dimensional space is not metrizable

Let $X $ be an infinite-dimensional normed space. I want to prove that the weak topology on $X$ is not metrizable. This is my solution: Assume that there is a metric $d$ on $X$ inducing the weak ...