All Questions
5,185 questions
2
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156
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Is there a better function (linear or even a projection)?
Let $A$ be a finite $n$-element set. Let $\mathbb R^A$ be an $n$-dimensional Euclidean space (with the ordinary Euclidean distance). Let $X$ be an arbitrary topological space. Consider a continuous ...
- 7,500
2
votes
0
answers
131
views
Topological dimension of quotient group determined by the inverse limit of discrete free monoids
Must the natural quotient group of the inverse limit of a sequence of nested discrete free monoids have topological dimension zero?
The question might well be open, but I would be grateful for news ...
- 1,968
2
votes
0
answers
248
views
A question about connected subsets of metric spaces
Let M be a metric space. Let T(M) be the topology of M (i.e. the collection of all open subsets of M)
and let C(M) be the collection of all connected subsets of M. In my opinion one often has a much
...
- 6,215
2
votes
0
answers
194
views
realcompact space
I want to study realcompact spaces but I can't find the best book or journal for it, and I really need to.
(sorry I don't write English very well)
- 21
2
votes
0
answers
272
views
Continuity of multiplicative character
Let $G$ be a discrete group and $\beta (G)$ denote the Stone-Cech compactification of $G$, a right topological semigroup. By a multiplicative character, I mean a mapping that preserves multiplication ...
- 61
2
votes
0
answers
371
views
Descriptive set theory on $\mathbb{R}^\mathbb{N}$
The short version of my question is, What is a good source for learning about descriptive set theory on the space $\mathbb{R}^\mathbb{N}$, under the product topology coming from the discrete topology ...
- 20.5k
2
votes
0
answers
1k
views
Closed irreducible subset
A nonempty subset $A$ of a topological space $X$ is called irreducible if, if $A\subset A_{1}\cup A_{2}$ and $A_{1}, A_{2}$ are closed subsets of $X$, then $A\subset A_{1}$ or $A\subset A_{2}$. We ...
- 192
2
votes
0
answers
192
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Are open convex PL subsets of R^n PL homeomorphic to R^n?
This is a basic issue of PL topology that I assume must be true, but I can't find a written reference: is a convex open PL subset of $\mathbb R^n$ PL homeomorphic to $\mathbb R^n$? I've scanned ...
- 5,544
2
votes
0
answers
146
views
How do you call a map which sends convergent sequences to pre-compact ones ?
In my work I encountered a map $f$ between two metric spaces $X$ and $Y$ that was not continuous (at least I couldn't prove it was), but I was able to prove that convergent sequences $(x_n)$ in $X$ ...
- 4,123
2
votes
0
answers
121
views
Graphs, non-Hausdorfness and Wallman compactifications of non-regular spaces
Given a non-Hausdorff space $X$, one can form a graph $G_X$: vertices the points of $X$, edges indicating point pairs not separated by open sets. Up to graph-theoretically (but not topologically) ...
- 17.6k
2
votes
0
answers
369
views
Constructing the Stone space of a distributive lattice
Does anyone have a good reference for the method of giving a topology to a distributive lattice as outlined in M.H. Stone's "Topological representation of distributive lattices and Brouwerian ...
- 10.5k
2
votes
0
answers
140
views
Products for probability theory using zero sets instead of open sets
(For all of this post, at least Countable Choice is assumed to hold.)
For all Tychonoff spaces $\langle X,\mathcal{T}\hspace{.06 in}\rangle$ :
Define $\mathbf{Z}(\langle X,\mathcal{T}\hspace{.06 in}\...
user5810
2
votes
0
answers
167
views
Local cartesian closedness in the category of compactly generated spaces
According the the nLab, the category of compactly generated (CG) spaces is not locally cartesian closed.
So if $A$ is a CG space and $C$ a CG space above $A$, $C$ may not be exponentiable.
What if we ...
- 3,033
2
votes
0
answers
1k
views
Double Torus Parametric Surface [closed]
In the process of trying to find continuous parametric surface equations for the double torus and for a pair of pants, I believe that the problem is unsolvable for some topological reason.
I have ...
2
votes
0
answers
565
views
Direct Limits and Limits of Nets
A net is a function from a directed set into a topological space, and it is said to converge to a point if certain conditions are satisfied. Similarly, a direct system is a function from a directed ...
- 15.4k
2
votes
0
answers
199
views
Finite topological dimension x local compactness
Of course, the two notions are independent one from the other, but often one of them implies the other under some additional hypotheses. For instance:
A topological vector space is finite dimensional ...
- 4,729
2
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0
answers
203
views
Faithful actions of finite groups on topological spaces
Suppose that $G$ is a finite group acting faithfully on a topological space $X$. In the smooth setting, one can deduce that for each $x$ in $M$, the induced map $$G_x \to Diff_x\left(M\right)$$ from ...
- 15.5k
2
votes
0
answers
254
views
Simple terminology question about the Dubrovnik (Kauffman) polynomial
In my S matrix classification attempts I encounter a lot of
Dubrovnik polynomials of the form D(z-1/z,z^n) and D(-z+1/z,z^n).
[Second variable is for writhe, n is an integer; for the first I don't
...
- 4,829
2
votes
0
answers
123
views
Constructing a lattice out of the set of metrics
Let $X$ be a space, and $d_1$ and $d_2$ be two metrics on $X$.
Define $S(x,y)= ${$\Sigma_2^l Min${$d_1(x_{k-1},x_k),d_2(x_{k-1},x_k)$}$:x_1=x, x_l=y, l finite $} $x$ and $y$ are two points in $X$
...
- 31
2
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0
answers
77
views
Characterizing local homeomorphisms into an exponent
Let $X$,$Y$, and $Z$ be (compactly generated) spaces. Suppose $f:Z \to Y^X$ is a local homeomorphism. How can we tell this from its adjoint $\tilde f:Z \times X \to Y$? I.e., I want a property $P$ ...
- 15.5k
2
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0
answers
185
views
Simple topological question on taking complements inside a simplex
We would like to know if the following claim is true:
(If you don't know the definition of a tropical hyperplane, then please consider the case when d=3)
Let $P_1,\cdots,P_d$ be full dimensional (...
- 113
2
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0
answers
299
views
Uniqueness of dimension for topological vector spaces
Let $V$ be a complete Hausdorff locally convex topological vector space over the field $\mathbb{K}$.
Let $B$ be a subset of $V$ satisfying
.
Linearly Independent: For all functions $f$ in $\mathbb{...
user5810
2
votes
0
answers
223
views
Is the realization of a proper map of simplicial spaces proper ?
Let $f:X \rightarrow Y$ be a map of $m$-dimensional simplicial spaces (which means that all simplices above dimension $m$ are degenerate). Recall, that $f$ is a natural transformation of functors from ...
- 11.1k
2
votes
0
answers
270
views
Homotopy equivalences and cores
Hi all,
Before asking my question, I need to fix some terms and notation.
Let $M$, $M'$ be locally compact, Hausdorff spaces, and $f:M\rightarrow M'$ a homotopy equivalence with homotopy inverse $g:...
- 93
2
votes
1
answer
118
views
Is every $b$-discrete space $X$ with countable injective weight basically disconnected?
Recall that a space $X$ is called basically disconnected [1] if every cozero-set has an open closure.
According to Tkačuk [2], a space $X$ said to be $b$-discrete if every countable subset of $X$ is ...
- 1,101
1
vote
4
answers
8k
views
Does Cauchy continuity imply uniform continuity? [No.] [closed]
It is well known that if $X$ is a first countable topological space and $Y$ is a topological space, then $f : X \rightarrow Y$ is continuous iff
$$\forall x \in {\rm map}(\mathbb{N},X),\forall p \in X ...
- 3,830
1
vote
3
answers
463
views
Openness of the set of injective functions in $C(\mathbb{R})$?
Let $C(\mathbb{R})$ be equipped with the topology of compact convergence (or equivalently the compact-open topology). Then, is the subset $\left\{f\in C(\mathbb{R}):
\text{$f$ injective}
\right\}$ ...
- 5,405
1
vote
3
answers
688
views
How to show the cardinality of nonisometric compact metric spaces is the continuum
It is asserted in A Course in Metric Geometry by Burago, Burago, Ivanov that
there can be no more than continuum of mutually nonisometric compact spaces
How is this proven?
Its clear that there ...
- 7,197
1
vote
3
answers
660
views
How can I construct a closed manifold from a finite CW complex?
If I start with a, say, 3-CW complex $X$ which can be embedded in $\mathbb{R}^5$, I can get a neighbourhood $U$ of $X$ which has the same homotopy type of $X$. Then $U$ is a $5-$ dimensional open ...
- 1,177
1
vote
2
answers
175
views
Non-homogeneous space $X$ such that $X\cong X\setminus \{x\}$ for all $x\in X$
What is an example of a topological space $(X,\tau)$ with the properties that
$X\cong X\setminus \{x\}$ for all $x\in X$, and
$(X,\tau)$ is not topologically homogeneous
?
- 48.9k
1
vote
1
answer
190
views
Is the semigroup M(n, Z) finitely presented? If so, where can I find a presentation of it?
I am new to semigroup research, so I apologize if this is an easy question.
- 87
1
vote
1
answer
1k
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A new generalisation of dimension? part 2
I worked this theory : A new generalization of the dimension?
I have a theorem about dimensions which is more general and simple than for matroids.
Definition 1: A structure $S$, is a pair $(X, \...
- 4,074
1
vote
2
answers
402
views
Homotopy problem for infinite dimensional topological space
Let $X$ be an infinite dimensional topological space such that :
$ \forall n \in \mathbb{N}$, $ \exists X_{n} \subset X$, $n$-dimensional subspaces verifying :
$\forall r<n$, the homotopy ...
1
vote
2
answers
1k
views
Possible errata in Nicolas Bourbaki's General Topology -I, Chapter 1 Exercise 2 ?
Here is the text of Exercise:
2 a) Let $X$ be an ordered set. Show that the set of intervals
$\left[x, \rightarrow\right[$ (resp. $\left]\leftarrow, x\right]$)
is a base of topology on $X$; ...
- 163
1
vote
2
answers
367
views
"Exactness" of groupify functor
For each commutative monoid $M$, there exists a "groupification" $\widehat{M}$, i.e. an abelian group that satisfies an obvious universal property.
I tried to prove the following: If in the diagram ...
- 9,853
1
vote
2
answers
484
views
Is there good evidence that topological spaces are the correct way to study the general theory of continuity? [closed]
My reason for asking is that the theory of metric spaces is so clean and so many significant theorems can be proved for an arbitrary metric space (which makes it plausible to me that metric spaces are ...
- 4,351
1
vote
2
answers
235
views
Can the Boolean Algebra of regular open sets be isomorphic to ${\cal P}(\omega)/(\text{fin})$?
Let $(X,\tau)$ be a topological space. $A\subseteq X$ is said to be regular open if $A = \text{int}(\text{cl}(A))$ and let $\text{RO}(X,\tau)$ denote the collection of regular open sets of $X$. A ...
- 48.9k
1
vote
3
answers
309
views
If $X$ is compact, is $[X]^2$ compact, too?
Let $(X,\tau)$ be a Hausdorff space. Let $[X]^2 = \big\{\{x,y\}: x,y\in X \land x\neq y\big\}$. For $U,V\in \tau$ with $U\cap V = \emptyset$ we set $[U,V] = \big\{\{x,y\} \in [X]^2: x\in U\land y\in V\...
user70876
1
vote
2
answers
304
views
Two notions of zero-dimensionality for topological spaces
Let $(X,\tau)$ be a topological space.
We say that $(X,\tau)$ is zero-dimensional with respect to the Lebesgue covering dimension (zd1) if every open cover of the space has a refinement which is a ...
- 48.9k
1
vote
2
answers
296
views
Are orbits of an affine algebraic monoid affine?
Let us work over the complex numbers for simplicity. Let $M$ be an affine algebraic monoid and $X$ an affine variety on which $M$ acts regularly, i.e. there is a morphism $\alpha: M\times X\to X$. Let ...
- 4,902
1
vote
3
answers
995
views
SO(3) knot polynomials
Can one use the real lie algebra so(3) to get knot polynomials? If so, do they have a skein relation (I presume they would, if they come from R-matrices in some standard way. If so, is the R-matrix ...
- 1,129
1
vote
3
answers
585
views
Terminology for certain monoids which are to monoids like fields are to rings
Let $M$ be a commutative monoid with zero. Then the condition $M^* = M \setminus \{0\}$ is very similar to the condition for a commutative ring to be a field. This analogy is also used in the work "...
1
vote
1
answer
424
views
Extension of homeomorphisms
Let $f,g:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be smooth injective and let $n\leq m$. Let $k \in \mathbb{N}$, and let $\iota_m^{m+k}:\mathbb{R}^m\rightarrow \mathbb{R}^{m+k}$ be the canonical ...
- 5,405
1
vote
2
answers
334
views
Connectedness of compact metric space
Let $X$ be a compact metric space satisfying the following condition: for any given positive number $\delta>0$, only finitely many components of $X$ have diameter larger than $\delta$.
For a given ...
- 97
1
vote
1
answer
132
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"Immovable" topological spaces
Let $(X,\tau)$ be a topological space. We define the "moving" relation by setting $$ x \simeq_m y \text{ iff there is a homemomorphism }\varphi: X\to X \text{ such that } \varphi(x) = y.$$
Clearly $\...
- 48.9k
1
vote
1
answer
202
views
Quotients of powers of the Sierpinski space
Is every space isomorphic to some quotient of a power of the Sierpinski space?
More precisely: Let $(X,\tau)$ be a topological space, and let $\mathbb{S} = (\{0,1\}, \{\emptyset, \{0\},\{0,1\})$ be ...
- 48.9k
1
vote
1
answer
1k
views
What is the meaning of non-Hausdorff spaces in algebraic geometry [closed]
At the beginning I should warn everybody reading this post: I don't know much about algebraic geometry so specialists in this subject may see my question as ignorant.
As far I understood one on the ...
- 9,340
1
vote
2
answers
171
views
Questions about knot (link) of surface in four dimension
Consider three 2-torus ($S^1*S^1$) living in four space. Can I have links of these objects, which is generalization of links of circles in 3D? If so, how can I judge whether three 2-torus are linked ...
- 293
1
vote
1
answer
479
views
Homology and homotopy of a surface
Suppose $S$ be a closed orientable genous $g$ surface. Let $f$,$g$ be homeomorphis from $S$ to itself. Assume they induce the same map on 1st homology $H_1(S, \mathbb Z).$
My question is; does this ...
user22741
1
vote
1
answer
400
views
$G_\delta$-diagonal
Could one find a counterexample that a topology space X is Tychonoff, seperable but hasn't
a $G_\delta$-diagonal? A topology space has a $G_\delta$-diagonal when there is a sequence
${G_n}$ of ...
- 654