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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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, $f(S)\subset Y$ ...
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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 $\...
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26 votes
1 answer
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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 ...
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26 votes
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Compact set of probability measures

I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
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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 ...
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21 votes
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437 views

Does every topological $n$-manifold ($n>0$) admit an embedding into $\Bbb R^{2n}$? If not, what $n$-manifold does not embed into $\Bbb R^{2n}$?

The strong Whitney Embedding Theorem tells us that every smooth n-manifold (n>0) admits a smooth embedding into $\mathbb{R}^{2n}$. Also, every topological $n$-manifold admits an embedding into $\...
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19 votes
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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 $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ ...
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19 votes
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611 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 ...
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18 votes
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254 views

Finite-Dimensional Homogeneous Contractible Spaces

Suppose that $X \subset \mathbb{R}^n$ is compact, homogeneous and contractible (and thus connected). Does $X$ have to be a point? I couldn't think of a non-trivial example, and there isn't a ...
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18 votes
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Stokes' Theorem general case

With the following lemma : Lemma : Let $f_{+},f_{-} : U \longmapsto \mathbb{R}$ be $C^{1}$ maps with $f_{-} \leq 0 \leq f_{+}$ with $U \subset \mathbb{R}^{n}$ open and bounded with $C^{1}$ boundary. ...
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16 votes
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Is there a topological space of cardinality $\kappa$ containing all metric spaces of cardinality $\kappa$?

It is known that all the countable metrizable spaces embed in $\mathbb{Q}$. Then a natural question is: are there an uncountable cardinal $\kappa$ and a topological space $X$ of cardinality $\kappa$ ...
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16 votes
2 answers
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Nerve Theorem: Is the finite union of closed convex sets triangulable?

My Question: Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable$^1$? If so, why? Background: I'm trying to better understand the ...
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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 $$\...
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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 ...
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14 votes
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Do we need full $\mathsf{AC}$ to efficiently use (sub)bases?

Suppose $(X,\tau)$ is a topological space, $B$ is a base for $\tau$, and $U\in \tau$ is an open set. Consider the following two strategies for writing $U$ as a union of elements of $B$: We have $U=\...
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14 votes
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When every point of a topological dynamical system is recurrent.

Definitions A topological dynamical system is a pair $(X, T)$ where $X$ is a compact metric space and let $T:X\to X$ is a continuous map. By the forward-orbit of a point $x$ in $X$ we mean the set $...
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14 votes
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262 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 $\...
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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)--for ...
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Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$?

Let $F\subset\Bbb{R} $ intersect every uncountable $\mathcal{F}_{\sigma}$ set. $B\subset \Bbb{R}$ is said to have the property of Baire if $B=U\triangle M$ where $U$ is open and $M$ is meager. Does ...
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Can ($X^I$, product topology) and ($X^I$, box topology) be homeomorphic for some nontrivial $X$ and infinite $I$?

Let $X$ be a nontrivial topological space, $I$ be a infinite set, we can endow $X^I$ (the set of all functions $I\to X$) with either the product topology or the box topology. We know that the box ...
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13 votes
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Forward characterization of measurable functions?

In topology, the standard definition of continuity works in the "backward" direction, since it puts a condition on the pre-images of a function rather than images: $f$ is continuous if the ...
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13 votes
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Adjunction of pointed maps is a homeomorphism?

What interests me the most is if the case of exponential law is true under the assumptions claimed for example on nlab: if $X, Y$ are Hausdorff and $Y$ is locally compact, then $F^0(X, F^0(Y, Z))\cong ...
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13 votes
1 answer
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Order-preserving map of regressive functions on $\omega_1$

I posted the following question in March 2014 on MO. It did receive some attention, but the answer there remains incomplete. It was motivated by some paracompactness-type properties as discussed at ...
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13 votes
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Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable. How does the following look? Proof: For each $n \in \mathbb{N}$, make an open cover of $K$ by ...
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13 votes
1 answer
528 views

Domain invariance for smooth functions

The domain invariance theorem states that for an open set $U\subset \mathbb{R}^n$ and a continuous and injective mapping $f:U\to \mathbb{R}^n,$ the image $f(U)\subset \mathbb{R}^n$ is open. I've read ...
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12 votes
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227 views

Want to study Graduate Measure Theory with heavy Emphasis on Topology and/or Geometry.

I did one course in Measure Theory and want to study it again. But this time I want to do this in a way that emphasizes Measure Theoretic structure on Geometric or Topological Spaces. I don't know, if ...
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11 votes
1 answer
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Space with semi-locally simply connected open subsets

A topological space $X$ is semi-locally simply connected if, for any $x\in X$, there exists an open neighbourhood $U$ of $x$ such that any loop in $U$ is homotopically equivalent to a constant one in $...
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11 votes
1 answer
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Topology on a space starting from topologies on subspaces

I have a question about constructing a topology on a space $X$ starting from topologies defined on a family of subspaces $(X_i)_{i\in I}$ of $X$. Assume that $X$ is a set and $(X_i)_{i\in I}$ is a ...
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11 votes
0 answers
265 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 ...
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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 ...
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11 votes
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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 ...
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11 votes
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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(...
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11 votes
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Conditions on $f$ such that separate continuity implies joint continuity

Consider a function $f : X\times Y \to Z$ where all spaces are compact metric spaces. Assume further that $f_x: y \mapsto f(x,y)$ and $f_y: x \mapsto f(x,y)$ are continuous. I am looking for ...
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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}(...
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11 votes
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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 ...
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10 votes
1 answer
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A topology over $\Bbb N$ based on convergence of series.

Define $\tau=\{U\subseteq \Bbb N:U\in\{\Bbb N,\emptyset\}\vee\sum_{n\notin U}n^{-1}<\infty\}$. In other words, a subset of $\Bbb N$ is closed iff it is $\Bbb N$ or the sum of the inverses of its ...
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10 votes
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Stone Duality: What are $\sigma$-Algebras Dual To?

Stone duality, one of many dualities between certain lattices and certain topological spaces, asserts that there is a contravariant categorical equivalence between the category $\text{Bool}$ of ...
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Find sequential orthographic projections, linking three different manifolds of dimension $n=1,2,3$

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. ...
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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 ...
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The topology of $GL(V)$

Let $V$ be a topological vector space (not necessarily finite-dimensional) over a field $K$, and let $GL(V)$ be the group of invertible linear maps $V\to V$ under composition. There are two obvious ...
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10 votes
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Prob. 1, Sec. 27, in Munkres' TOPOLOGY, 2nd ed: How to show that the compactness of every closed interval implies the least upper bound property?

Here is Prob. 1, Sec. 27, in the book Topology by James R. Munkres, 2nd edition: Prove that if $X$ is an ordered set in which every closed interval is compact, then $X$ has the least upper bound ...
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10 votes
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242 views

Elementary proof of compact space = exhaustible space?

(This is a repost of a question I asked last year on cs.stackexchange.) The work of Martín Escardó has demonstrated close parallels between classical topology on one hand and computability on the ...
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10 votes
0 answers
220 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 ...
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10 votes
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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 ...
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10 votes
0 answers
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 ...
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10 votes
1 answer
207 views

When is $\{ x | f(x) \le 0\}$ path-connected?

I'm trying to determine the conditions on $f : \mathbb{R}^n_{\ge 0} \to \mathbb{R^n}$ under which $\{ x | f(x) \le 0\}$ is path-connected. We can assume that $f$ is continuous and concave (i.e. for ...
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  • 4,026
9 votes
0 answers
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Infinite (co)-homology

Lately, I've been wondering if it was possible to define singular homology also with infinite-dimensional simplices. For example we could define an infinite dimensional simplex as: $$\Delta_{\infty}:=\...
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9 votes
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Is this an alternative definition of manifolds?

Suppose $X\subseteq\mathbb{R}^m$ s.t. for any $x\in X$ and any open $U\subseteq\mathbb{R}^m$ that contains $x$, there exists a smaller open set $V\subseteq U$ also containing $x$, so that $V\cap X$ is ...
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9 votes
0 answers
83 views

Understanding Brouwer Separation Theorem with an easier proof

I'd like to undersand better Brouwer separation theorem given in Massey (Proposition 6.5 p.215) since has some smoky parts to me. To lighten the notation we set $D^n := \mathbb{D}^n, S^n := \mathbb{S}^...
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9 votes
0 answers
183 views

Name/source for cardinal invariant in topology

Given a topological space $\mathcal{X}=(X,\tau)$ and a set $A\subseteq X$, say that $A$ is $\mathcal{X}$-sufficient iff every continuous function $(A,\tau_A)\rightarrow\mathbb{R}$ extends to a unique ...
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