Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [gn.general-topology]

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

2
votes
0answers
201 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 ...
246
votes
35answers
60k views

Why is a topology made up of 'open' sets? [closed]

I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...
19
votes
2answers
1k views

Colimits in the category of smooth manifolds

In the category of smooth real manifolds, do all small colimits exist? In other words, is this category small-cocomplete? I can see that computing push-outs in the category of topological spaces of ...
2
votes
0answers
263 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:...
3
votes
3answers
661 views

Is there a name for this property of a topology?

This property seems like it should have a nice name, but I can't find one anywhere. Does anyone know a name for this? For each non-empty open set $U$, there exist proper open subsets $\{U_i\}_{i\in ...
2
votes
2answers
1k views

Is a proper quotient map closed ?

I am trying to produce closed quotient maps, as they allow a good way of creating saturated open sets (as in this question). A map $f:X\rightarrow Y$is called proper, iff preimages of compact sets ...
0
votes
3answers
237 views

how slow can the dimension of a product set grow?

Let us define the following "dimension" of a Borel subet $B \subset \mathbb{R}^k$: $\dim(B) = \min\{n \in \mathbb{N}: \exists K \subset \mathbb{R}^n, ~{\rm s.t.} ~ B \sim K\}$, where $\sim$ denotes "...
10
votes
2answers
2k views

Continuous function from $[0,1]$ to $[0,1]$

Does there exist a continuous function $f:[0,1]\rightarrow [0,1]$ such that $f$ takes every value in $[0,1]$ an infinite number of times?
15
votes
3answers
3k views

Is a inverse limit of compact spaces again compact ?

Then one can construct a model for the inverse limit by taking all the compatible sequences. This is a subspace of a product of compact spaces. This product is compact by Tychonoff. If all the spaces ...
9
votes
1answer
615 views

What is enough to conclude that something is a CW complex?

This question was something I considered when looking into CW-structures on Grassmannians, but I found no general treatment of this in the literature: Question: Assume that $X$ is an $n-1$ ...
3
votes
2answers
497 views

SU(2) representations of alternating knot groups

Suppose that $K$ is an $\textit{alternating}$ knot in $S^3$, and let $R_0$ be the space of homomorphisms from $\pi_1(S^3 - K)\to SU(2)$ which send meridians to trace free matrices. Denote the subset ...
9
votes
3answers
751 views

Is there a non-trivial topological group structure of $\mathbb{Z}$?

More specificaly, is there a haussdorf non-discrete topology on $\mathbb{Z}$ that makes it a topological group with the usual addition operation?
11
votes
2answers
1k views

Is this a known compactification of the natural numbers?

Given two infinite sets $A$, $B$ of natural numbers, write $A\preceq B$ if $B\setminus A$ is a finite set. Define the equivalence relation $A\sim B$ if $A\preceq B$ and $B\preceq A$, and let $\partial\...
3
votes
2answers
2k views

What do you call a topology that is closed under arbitrary intersections?

An arbitrary union, or a finite intersection, of open sets in a topological space is again open. What name is given to the hypothetical property that an arbitrary intersection of open sets is open? ...
7
votes
2answers
346 views

Relation between $KO$ and $K$

What can be said about the relation between the complex and the real K-theory of a CW complex? An $n$-dimensional complex vector bundle is an $2n$-dimensional real vector bundle but not vice versa. ...
1
vote
2answers
584 views

Existence of convergent subsequences for all values in range?

Consider sequence $s(n) = \sin{nx}$. Are there values of $x$ for which the following holds: For every $y \in \[-1,1\]$ there is a subsequence of $s(n)$ converging to $y$? (Or perhaps just for the open ...
29
votes
5answers
2k views

Is the universal covering of an open subset of $\mathbb{R}^n$ diffeomorphic to an open subset of $\mathbb{R}^n$ ?

Is the universal covering of a connected open subset $U$ of ℝn diffeomorphic to an open subset of ℝn (standard differentiable structure)? If not true in general, is there any condition ...
4
votes
2answers
620 views

References and applications involving the Krull Toplogy

I was wondering if anyone can suggest a reference which treats the Krull topology. Most of the books I have found don't go into any kind of detail. It is my understanding that the Krull topology ...
26
votes
5answers
3k views

Two-to-one continuous mapping from R² to R²

Hello. I have a question. Does there exist a continuous mapping $F:\mathbb{R}^2\rightarrow\mathbb{R}^2$ such that for every $c\in F(\mathbb{R}^2)$ there are two and only two points $z_{1}$, $z_{2}$...
87
votes
9answers
27k views

solving $f(f(x))=g(x)$

This question is of course inspired by the question How to solve f(f(x))=cosx and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a ...
2
votes
1answer
927 views

Finding saturated open sets

Suppose I have a continuous map $f:X\rightarrow Y$. Then one can wonder, whether for every open set $U\subset X$ the set $U':=\{x\in X|f^{-1}(f(x))\subset U\}$ is open again. This is not true in ...
10
votes
2answers
299 views

existence of a connected set with given connected projections.

Suppose A and B are compact connected sets in the XY plane and XZ plane respectively in R^3. Suppose further that the the range of x-values taken by A and B are the same (i.e, projections of A and B ...
11
votes
4answers
3k views

Regular spaces that are not completely regular

In the undergraduate toplogy course we were given examples of spaces that are $T_i$ but not $T_{i+1}$ for $i=0,\ldots,4$. However, no example of a space which is $T_3$ but not $T_{3.5}$ was given. ...
5
votes
4answers
797 views

Fundamental domains of measure preserving actions

Suppose a finite group $G$ acts on a standard probability space $(X, \mu)$ by measure-preserving actions (i.e. $\mu(g(A)) = \mu(A)$ for all $g \in G$ and $A \subset X$ measurable). In addition, ...
3
votes
1answer
1k views

Fiber bundle = principal bundle + fiber?

This question is heavily related to this question. Fix a sufficiently nice and connected topological space $B$ and let $FB$ be the category of fiber bundles over $B$. A morphism $f: (E\to B)\to (E'\...
4
votes
4answers
1k views

What is a reference for profinite sets?

The question is in the title. The motivation behind the question is as follows: there are plenty of references about profinite groups and profinite completions of groups. It seems that their not ...
6
votes
3answers
1k views

Using topology to characterize embedded Lie subgroups of Lie groups.

Cartan's theorem states that any topologically closed subgroup of a Lie group is an embedded Lie subgroup. This leads us to ask the following question: Can we replace "topologically closed" with a ...
30
votes
3answers
1k views

Can a connected planar compactum minus a point be totally disconnected?

What the title said. In a slightly more leisurely fashion:- Let $X$ be a compact, connected subset of $\mathbb{R}^2$ with more than one point, and let $x\in X$. Can $X\smallsetminus\{x\}$ be ...
29
votes
7answers
4k views

Why is it useful to classify the vector bundles of a space?

It seems to me that vector bundles are useful because they allow us to bring to bear all of the linear algebra we know to aid in the study of topological spaces. Now, I've read somewhere that it is ...
7
votes
2answers
1k views

Intersection form in twisted homology (homology with local coefficients)

The answer to this question should be obvious, but I can't seem to figure it out. Suppose we have a surface $F$, and a representation $\rho : \pi_1(F)\to SU(n)$. We can define the homology with local ...
15
votes
1answer
2k views

Which Fréchet manifolds have a smooth partition of unity?

A classical theorem is saying that every smooth, finite-dimensional manifold has a smooth partition of unity. My question is: Which Fréchet manifolds have a smooth partition of unity? How is the ...
67
votes
5answers
4k views

How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?

This question is about the space of all topologies on a fixed set X. We may order the topologies by refinement, so that τ ≤ σ just in case every τ open set is open in σ. ...
11
votes
5answers
1k views

Confusion over a point in basic category theory

"Let Top be the category of topological spaces." If I see a definition like this, in which homeomorphic (isomorphic in the category) spaces are not identified together, then for each given topological ...
21
votes
8answers
2k views

Cryptomorphisms

I am curious to collect examples of equivalent axiomatizations of mathematical structures. The two examples that I have in mind are Topological Spaces. These can be defined in terms of open sets, ...
4
votes
1answer
329 views

“Category” of Nonempty Metric Spaces and Contractive Maps?

The usual way of getting a category of metric spaces is to take metric spaces as objects, and the nonexpansive maps (ie, functions $f : A \to B$ such that $d_B(f(a), f(a')) \leq d_A(a, a')$) as ...
1
vote
2answers
844 views

Simple question of topological cofibration

I have an inclusion of topological spaces (actually manifolds with corners) $X \to Y$. I can show that for every $x \in X$ there is a neighborhood of $x$ in $Y$ of the form $U \times V$. Also, the ...
25
votes
4answers
3k views

Which spaces are inverse limits of discrete spaces ?

There is the following theorem: "A space $X$ is the inverse limit of a system of discrete finite spaces, if and only if $X$ is totally disconnected, compact and Hausdorff." A finite discrete space ...
2
votes
1answer
315 views

Topologies making a class of functions continuous [closed]

Let $X:=\{f: \mathbb{C}\to \mathbb{C}\}$ be a class of total functions on $\mathbb{C}$ closed under composition, addition, multiplication, and scalar multiplication. Does there exist a topology on $\...
6
votes
3answers
349 views

Notion of finite dimensional simplicial space

I was wondering, what a $N$-dimensional simplicial space $X$ should be. Of course the degeneracy maps force the spaces to be nonempty in high dimensions. Currently I have two different versions and i ...
37
votes
7answers
4k views

Why should algebraic objects have naturally associated topological spaces? (Formerly: What is a topological space?)

In this question, Harry Gindi states: The fact that a commutative ring has a natural topological space associated with it is a really interesting coincidence. Moreover, in the answers, Pete L. ...
6
votes
3answers
892 views

The continuous as the limit of the discrete

Reading this documment: www.math.ucla.edu/~tao/preprints/compactness.pdf, I got interested in the following thing: "One can also use compactifications to view the continuous as the limit of the ...
44
votes
7answers
6k views

“Algebraic” topologies like the Zariski topology?

The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact. ...
-4
votes
2answers
1k views

Finite versus infinite on non-Hausdorff topologies [closed]

Question: Does there exist some real-valued function $f(x)$ where $f: \mathbb{R} \to \mathbb{R}$, for which $\lim_{x \to \infty}$ converges on a non-Hausdorff topology but does not converge on a ...
13
votes
2answers
541 views

Functions separting points in Hausdorff spaces

A colleague in algebra asked me this, and I couldn't answer it. On the Wikipedia page for "epimorphism" it is claimed that in the category of Hausdorff spaces and continuous maps, a function is epi ...
4
votes
1answer
492 views

Sheaf condition and representability in the category Top

This is a rather nice question I got from this user via private communication. Let $\mathcal{C} = Top$ the category of topological spaces. Let $\mathcal{C}^\prime$ be the category $Funct(\mathcal{C}^{...
2
votes
6answers
2k views

Cone in a metric space

Hi everybody, We know the definition of a cone in a Real Banach Space. I want to know if there is any definition for a cone in an abstract metric space. Have you ever seen such definition anywhere? ...
4
votes
1answer
2k views

definition of the end of a manifold?

Hey everybody, I was hoping if somebody could help me out with the terminology. I've found that the "end of a manifold" is a function asigning to each compact set K a conected component e(K) of the ...
2
votes
3answers
1k views

Baire category theorem

Let's call the following conditions (1): $X$ is a complete metric space with metric $d$, $X = \cup_{n=1}^\infty A_n$. Let $\bar{A}$ denote the closure of $A$. Let's call the following statement (2): ...
17
votes
4answers
2k views

What is the “right” universal property of the completion of a metric space?

I'm a little embarrassed to ask this one, but it could help for a class I'm teaching, so here goes: Let $X$ be a metric space. We all know that $X$ admits a completion, which is a complete metric ...