# Questions tagged [gn.general-topology]

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

**2**

votes

**0**answers

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

**35**answers

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

**2**answers

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

**0**answers

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

**3**answers

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

**2**answers

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

**3**answers

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

**2**answers

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

**3**answers

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

**1**answer

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

**2**answers

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

**3**answers

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

**2**answers

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

**2**answers

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

**2**answers

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

**2**answers

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

**5**answers

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

**2**answers

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

**5**answers

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

**9**answers

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

**1**answer

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

**2**answers

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

**4**answers

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

**4**answers

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

**1**answer

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

**4**answers

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

**3**answers

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

**3**answers

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

**7**answers

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

**2**answers

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

**1**answer

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

**5**answers

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

**5**answers

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

**8**answers

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

**1**answer

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

**2**answers

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

**4**answers

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

**1**answer

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

**3**answers

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

**7**answers

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

**3**answers

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

**7**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**6**answers

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

**1**answer

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

**3**answers

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

**4**answers

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 ...