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.
9,729
questions with no upvoted or accepted answers
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Is there 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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44
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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 $\...
user160185
30
votes
1
answer
858
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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 ...
- 5,256
29
votes
1
answer
3k
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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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28
votes
0
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673
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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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23
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0
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879
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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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23
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0
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339
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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 ...
- 293
21
votes
0
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617
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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. ...
- 5,174
20
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0
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536
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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 $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ ...
- 10.5k
19
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0
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338
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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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19
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0
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367
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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 $\kappa$ ...
- 2,073
19
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0
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656
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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 ...
- 9,354
17
votes
1
answer
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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 ...
- 13.1k
17
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0
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343
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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
$$\...
- 6,840
17
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0
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372
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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 ...
- 54.5k
17
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0
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503
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Homeomorphism between evenly spaced integer topology and the rationals
The evenly spaced integer topology is countable, metrizable, and has no isolated points, and hence is homeomorphic to the rationals with the order topology. But what is an explicit construction for ...
- 8,210
16
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0
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458
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Is the set of points in $\mathbb R^3$ with exactly one rational coordinate connected?
Let
\begin{align}
S&=\{(x,y,z)\in\mathbb R^3:\text{exactly one of }x,y,z\text{ is rational}\}\\
&=\mathbb Q^c\times\mathbb Q^c\times\mathbb Q\cup \mathbb Q^c\times\mathbb Q\times\mathbb Q^c\...
- 1,803
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0
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261
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Do we need full $\mathsf{AC}$ to efficiently use (sub)bases?
Very belatedly, sorry: also asked at MO.
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 ...
- 251k
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1
answer
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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 ...
- 3,289
15
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0
answers
313
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Is a function $\mathbb{R}^n \to \mathbb{R}$ which has a closed and connected graph necessarily continuous?
It has been proved here and here that a function
$\mathbb{R} \to \mathbb{R}$ which has a closed and connected graph is continuous.
This fact is also proved in a nice article by Burgess. I don't know ...
- 1,504
15
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0
answers
279
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 $\...
- 3,538
15
votes
0
answers
585
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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 ...
- 104k
14
votes
0
answers
590
views
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 $...
- 18.8k
14
votes
1
answer
736
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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13
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0
answers
427
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Any reference on Jensen inequality for measurable convex functions on a Banach space?
The only proof of Jensen inequality (and most general version) that I know is a direct consequence of the Fenchel-Moreau Theorem : If $X$ is a locally convex Hausdorff topological space, let $\mu$ be ...
- 5,967
13
votes
0
answers
299
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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 ...
- 6,804
13
votes
0
answers
602
views
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 ...
- 10.8k
13
votes
0
answers
948
views
Topologist's sine curve is a simply-connected space
I am trying to solve the following problem from Hatcher's Algebraic Topology and have written a solution. Could you help me checking my solution, whether I am right? Thanks in advance.
$Y$ is simply-...
- 9,654
13
votes
2
answers
412
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 ...
- 3,547
13
votes
0
answers
636
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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 ...
- 18.5k
13
votes
1
answer
550
views
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 ...
- 13.6k
13
votes
1
answer
327
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 ...
- 1,590
12
votes
0
answers
198
views
Are neighbourhood deformation retracts transitive?
I have the following definition of a neighbourhood deformation retract (or NDR, for short): A pair $(X, A)$ is called an NDR if $A\subseteq X$ is closed and there exists a neighbourhood $A\subseteq V\...
- 2,029
12
votes
0
answers
344
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Construction of the Universal Covering Space via Compact-Open Topology
Recently I've been self-studying the theory of covering spaces from "Introduction to Topological Manifolds", by John M. Lee. At the end of Chapter 11, there is an explicit construction of ...
- 1,652
12
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0
answers
386
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 ...
- 407
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462
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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 ...
- 3,575
12
votes
0
answers
724
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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(...
- 24k
11
votes
0
answers
279
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Is there a path-connected, "anti-convex" subset of $\mathbb R^2$ containing $(\mathbb R\smallsetminus \mathbb Q)^2$?
This question was firstly asked in Mathematics Stack Exchange. Getting no answer, I copied it to Math Overflow, as Moishe Kohan commented.
For a vector space $V$ over $\mathbb R$, I say a subset $S$ ...
- 358
11
votes
0
answers
192
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A natural topology on a field
I can endow any field with a natural topology in the following way. Given a polynomial $f\in K[X]$, I denote by $\mathcal{O}(f)=\{x\in K\mid\exists y\in K^{\times}\ f(x)=y^2\}$, i.e. the set of ...
- 329
11
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0
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239
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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}:=\...
- 3,264
11
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0
answers
145
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Topology on power set such that union is continuous
Let $C$ be a set and let $2^C$ be its power set. Consider the union function $\cup: 2^C\times 2^C\rightarrow 2^C$ such that $\cup (X,Y)=X\cup Y$. For a fixed $Y\in 2^C$, let $\cup_Y$ be the evaluation ...
11
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0
answers
453
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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 ...
user900250
11
votes
1
answer
546
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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 $...
- 621
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votes
1
answer
129
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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 ...
- 131
11
votes
0
answers
149
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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 ...
- 438
11
votes
0
answers
855
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 ...
user290300
11
votes
0
answers
972
views
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 ...
- 27.8k
11
votes
0
answers
2k
views
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 ...
- 797
11
votes
0
answers
354
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}(...
- 1,437
11
votes
0
answers
1k
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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 ...
- 54.5k