Questions tagged [gn.general-topology]

Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

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Filling $\mathbb{R}^3$ with skew lines

I would like to know if it is possible to fill $\mathbb{R}^3$ with lines with the following two properties: (1) Every point $x \in \mathbb{R}^3$ is contained in precisely one line. (2) Every ...
Joseph O'Rourke's user avatar
7 votes
0 answers
512 views

Is there a natural topology on the set of open sets ?

Given a topological space $(X,\mathcal{O})$ can one assign a natural topology to $\mathcal{O}$ such that 1) The intersection of a compact set of open sets is again open, 2) The maps $\cap,\cup:\...
HenrikRüping's user avatar
6 votes
2 answers
255 views

Borel functions on $\omega_1$

Endow $\omega_1$ with order topology. It is easy to show that each continuous function $f\colon \omega_1\to \mathbb{R}$ is eventually constant. Is the same true for Borel functions?
Kulikov's user avatar
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49 votes
7 answers
5k views

Is there an algebraic approach to metric spaces?

It is well known that most topological spaces can be studied via their algebra of continuous real-valued (or complex-valued) functions. For instance, in the setting of compact Hausdorff spaces, there ...
Mark's user avatar
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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) ...
David Feldman's user avatar
4 votes
1 answer
310 views

General topology terminology questions

In a Hausdorff but not regular space, collapsing certain closed sets to a point may produce a non-Hausdorff space. Does there exist a term for closed sets one may collapse and still have a Hausdorff ...
David Feldman's user avatar
2 votes
1 answer
213 views

union of Stone-Cech remainders

Can anyone point me to a reference or further information on the following construction? Let $X$ be a compact metric space such as $[0,1]$. Let $A$ be the commutative pre-C*-algebra consisting of [...
Douglas Somerset's user avatar
13 votes
1 answer
537 views

Square of a continuous map

Recently a student asked me the following (elementary looking) question : If $T$ is an invertible linear transformation of some finite-dimensional space $E$ into itself which factorizes as $T = f \...
js21's user avatar
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20 votes
1 answer
955 views

Which spaces are characterized by functions with compact support ?

It's well known that two locally compact Hausdorff spaces $X, Y$ are homeomorphic iff the rings $C_0(X), C_0(Y)$ (continuous functions vanishing at infinity) are isomorphic. Is there a class $\...
Ralph's user avatar
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4 votes
1 answer
215 views

closed set and z-ultrafilter on normal space

Let $X$ be a completely regular, Hausdorff topological space and let $\cal F$ be a $z$-ultrafilter on $X$. Then for each zero set $W$ in $X$, either $W\in \cal F$ or there exists $Z\in \cal F$ such ...
Douglas Somerset's user avatar
9 votes
3 answers
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Where can I find a proof of the de Rham-Weil theorem?

Where can I find a proof of the de Rham-Weil theorem? Does anyone know?
Louis A's user avatar
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23 votes
4 answers
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Spaces with no topological monoid structure which are homotopy equivalent to topological monoids

In motivating $A_\infty$-spaces to my students I'm going to insist on the homotopy invariance of the notion, saying that "being $A_\infty$ is the homotopy invariant version of being a topological ...
domenico fiorenza's user avatar
36 votes
2 answers
3k views

Computing self-intersections with complex analysis

It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis: $$n = \oint_C\frac{dz}{z}.$$ You can also count the number of roots of $f(z) = 0$ inside a close ...
john mangual's user avatar
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5 votes
1 answer
445 views

Least cardinality of a set of points in the plane

What is the least possible cardinality $K$, of a set S of points in the plane, such that there exists a point P in the plane and an open ball B centered at P, such that for all points X in B, not all ...
Holowitz's user avatar
6 votes
1 answer
390 views

Infinite closed partition of the real line with no closed infinite unions

Is there a partition of the real line into infinitely many closed subsets so that no infinite union of these subsets (except the whole space) is closed? This question was asked also at math....
LostInMath's user avatar
9 votes
2 answers
2k views

Stone-Čech compactification of $\mathbb R$

Let $\beta X$ - is a Stone-Čech compactification of $X$. $I=(-1,1)$ - is an interval of the real line. Is it true that $\beta \mathbb R\setminus I = \beta(\mathbb R\setminus I)$? In other words, it ...
Mariarty's user avatar
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11 votes
1 answer
1k views

Reference request for TQFT, functoriality

I am reading Turaev's blue book Quantum Invariants of Knots and 3-manifolds. It is difficult for me to understand the proof of Theorem 1.9 in chapter 4, which says; The function $(M, \partial_{-}M, \...
Link's user avatar
  • 111
3 votes
1 answer
514 views

Metrizable implies hemicompact

In the paper R. Arens: A Topology for Spaces of Transformations, Ann. of Math. 47(1946), 480-495 the author states in the introduction that if $B$ is a metric space and the space of continuous ...
Ralph's user avatar
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4 votes
2 answers
731 views

Moore path space.

Let $X$ a topological space and $MX$ the Moore path space of $X$ there is two maps from $\alpha,\omega: MX\rightarrow X$ (evaluation in 0 and evaluation at the total length). The classical path ...
Ilias A.'s user avatar
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6 votes
0 answers
690 views

What is the structure of a space of $\sigma$-algebras?

Let $X$ be a compact metric space, and consider the Banach space $\Omega = C(X,\mathbb R)$ of continuous, real-valued functions on $X$, equipped with the supremum norm. Let $\delta_x \in \Omega^*$ be ...
Tom LaGatta's user avatar
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2 votes
2 answers
682 views

Topology on Set of Prime Filters of a Distributive Lattice

Given a distributive lattice $A$ we can look at $Spec(A)$, whose points are prime ideals and its open sets are given similarly to the Zariski topology on Spec of a ring. That is, the basis of open ...
Jonathan Beardsley's user avatar
5 votes
0 answers
131 views

Possible homogeneity of infinite dimensional Sierpinski carpet analogues?

Start with the Hilbert cube $H=I^\omega$, thinking of its coordinates as written in ternary expansion. Construct subsets $S_n$ by removing points from $H$ if for any $m$, at least $n$ of the ...
David Feldman's user avatar
6 votes
1 answer
759 views

When is a Topological pushout also a Smooth pushout?

I feel like this problem has not been solved, but I'm interested in knowing any results on it. More specifically, I mean: Let $B\stackrel{f}{\leftarrow} A \stackrel{g}{\rightarrow} C$ be a diagram ...
William's user avatar
  • 712
4 votes
1 answer
2k views

Connected level sets

This may be an ill-posed question, but suppose I have a collection of continuous, bounded, scalar-valued nonnegative functions $f_1(x,y),\dots,f_n(x.y)$ defined on the closed unit disk. Given a ...
Jennifer Gao's user avatar
4 votes
0 answers
223 views

A closure operation on subsets of ${\Bbb Z}[x]$

Given a(n infinite) set $S\subset {\Bbb Z}[x]$ (integer polynomials), write $R_S$ for the topological closure of the set of all complex roots of all $p\in S$. Then write $\hat{S}$ for the set of all ...
David Feldman's user avatar
18 votes
4 answers
3k views

When is a finite cw-complex a compact topological manifold?

I think the statement of the question is pretty straightforward. Given a finite $n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact $n$-...
William's user avatar
  • 712
0 votes
0 answers
544 views

Visualizing self-homeomorphism of a cylinder over a torus

A cylinder over a torus is by definition $S^1 \times S^1 \times I$ , here $I=[0,1]$. One way to visualize it is to thick a torus in $\mathbb{R}^3$. ( $S^1 \times I$ is an annulus, and revolve it (...
knot's user avatar
  • 93
5 votes
1 answer
363 views

Ring of a Spectral Space

It is said, as far as I can tell that an arbitrary spectral space, i.e. a space that is $T_0$, sober and quasi-compact whose collection of quasi-compact open sets forms a basis and is closed under ...
Jonathan Beardsley's user avatar
2 votes
1 answer
402 views

Automorphism of first homology and mapping class group

It is known that for a torus $\Sigma$, every automorphism of $H_1(\Sigma; \mathbb{Z})$ is induce by an orientation preserving self-homeomorphism of $\Sigma$ unique up to isotopy. In onther words, ...
knot's user avatar
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1 vote
1 answer
311 views

Mapping class group and cylindrical structure

Let us fix a torus $\Sigma=S^1 \times S^1$. We consider a cylinder $\Sigma \times I$ and a data $(\Sigma\times I, \Sigma\times 0, \Sigma\times 1, f_{0},f_{1})$. Here $f_{i}$, called parametrization, ...
knot's user avatar
  • 93
18 votes
2 answers
2k views

Example of a weak Hausdorff space that is not Hausdorff?

I've looked on the web and haven't found a simple example.
Bob Solovay's user avatar
11 votes
5 answers
5k views

A criterion for the sum of two closed sets to be closed ?

Let $V$ and $I$ be two closed subsets of a Banach space $A$. The set $V$ is a convex cone, and $I$ is a linear subspace of $A$. I also know that $V\cap I=\{0\}$. I would like to know whether $I+V$ ...
Fabien Besnard's user avatar
2 votes
0 answers
343 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 logics"? ...
Jonathan Beardsley's user avatar
1 vote
1 answer
2k views

Covering the Rationals -- A Paradox? [closed]

Covering the Rationals -- A Paradox? The following seems to yield a paradox. Can anyone provide the proper resolution? Preamble It is easy to show that between any two reals there is a rational. If ...
Ashley McNeile's user avatar
110 votes
4 answers
13k views

Is there a sheaf theoretical characterization of a differentiable manifold?

I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...
Daniel Moskovich's user avatar
10 votes
1 answer
833 views

Completeness of Borel measure

Let $X$ be a compact Hausdorff space and $\mu$ a finite Borel measure without atoms which is outer regular with respect to open sets and inner regular with respect to compact sets. Can such measure be ...
arc's user avatar
  • 277
10 votes
3 answers
1k views

Is the reals the smallest connected ordered topological ring?

The real numbers is a locally compact Tychonoff connected complete ordered topological field. I am looking at minimal collections of adjectives that can characterize the reals. The one often used to ...
user avatar
5 votes
1 answer
721 views

Characteristic classes of a fibered sum

I will phrase this question in terms of attaching smooth manifolds along a submanifold, though it is certainly more general. Let $M_1$ and $M_2$ be smooth $n$-manifolds (maybe closed, for simplicity),...
William's user avatar
  • 712
90 votes
3 answers
13k views

Is every sigma-algebra the Borel algebra of a topology?

This question arises from the excellent question posed on math.SE by Salvo Tringali, namely, Correspondence between Borel algebras and topology. Since the question was not answered there after some ...
Joel David Hamkins's user avatar
45 votes
2 answers
5k views

Continuous bijections vs. Homeomorphisms

This is motivated by an old question of Henno Brandsma. Two topological spaces $X$ and $Y$ are said to be bijectively related, if there exist continuous bijections $f:X \to Y$ and $g:Y \to X$. Let´s ...
Ramiro de la Vega's user avatar
103 votes
5 answers
16k views

Independent evidence for the classification of topological 4-manifolds?

Is there any evidence for the classification of topological 4-manifolds, aside from Freedman's 1982 paper "The topology of four-dimensional manifolds", Journal of Differential Geometry 17(3) 357–453? ...
Brendan Guilfoyle's user avatar
-1 votes
1 answer
538 views

Fuzzy topology : references [closed]

Hey. I'm looking for references in fuzzy topology. Does anyone know a good book ?
Dimitri's user avatar
  • 11
5 votes
3 answers
668 views

Does every compact Hausdorff ring admit a decomposition into primitive idempotents?

Let $\mathbf{R} = (R,\mathcal{T},+,\cdot,0,1)$ be a compact Hausdorff topological unitary ring, and consider the set $I(\mathbf{R}) := \{ e \in R \mid e \cdot e = e \}$ (of idempotents in $\mathbf{R}$)...
Niemi's user avatar
  • 1,488
1 vote
0 answers
199 views

Soft sheaves on indiscrete paracompact spaces

Let $X$ be some space, I have basically 2 questions: 1 - Are sheaves on paracompact but not Hausdorff spaces acyclic? I've been doing some reading and some authors say that soft sheaves on ...
Richard Jennings's user avatar
13 votes
2 answers
2k views

Well-pointed space which is not locally contractible

I am looking for an example of a well-pointed space in which no (sufficiently small) neighbourhood of the base-point is contractible. As usual, a well-pointed space is a pointed space in which the ...
Ricardo Andrade's user avatar
6 votes
0 answers
558 views

Continuous images of Cantor cubes

The original title of this question was "Is there only one (up to homeomorphism) zero-dimensional homogeneous dyadic space of weight $\mathfrak{c}$?". I changed it with the hope of getting a bit more ...
Ramiro de la Vega's user avatar
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}\...
user avatar
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 $X$; ...
nature1729's user avatar
2 votes
3 answers
561 views

A simple question on the closure of the image of a morphism

Let $X$ be a complex irreducible quasi-projective variety, $f:X\longrightarrow\mathbb{P}^N$ a morphism, $H\subset\mathbb{P}^N$ a hyperplane, $Z:=f^{-1}(H)$ which is irreducible, $Y\subset X$ a ...
gio's user avatar
  • 1,149
3 votes
2 answers
679 views

Separation axioms

Reading about separation axioms, I wonder: Is there a separation axiom weaker than $T_2$ but stronger than $T_1$? $T_{1.5}$? I suppose there are some separation axioms stronger that $T_6$, how many ...
Pedro Perez's user avatar

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