# Questions tagged [gn.general-topology]

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

**5**

votes

**2**answers

525 views

### When does Scott topology generated by specialization order induced by a sober space (X,$\tau$) equal the initial topology $\tau$?

Let X be a $T_{0}$ space. The specialization order ≤ on X is that if x is contained in cl{y}, then we call "x≤y". Obviously (X,≤) is a partially ordered set.
A sober space is a topological space such ...

**34**

votes

**1**answer

2k 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 ...

**7**

votes

**1**answer

175 views

### How much can complexities of bases of a “simple” space vary?

Given a countable subbase of a topology, we can consider its complexity in terms of the difficulty of determining whether one family of basic open sets covers another basic open set. My question is ...

**5**

votes

**1**answer

253 views

### Topology of connected subsets of the $3$-torus

Consider the $3$-torus $Y=T^3$, a subset $\Sigma\subset Y$, and $\Sigma^*=Y\setminus\overline\Sigma$.
We assume both $\Sigma$ and $\Sigma^*$ to be open, connected, and smoothly bounded.
I am ...

**5**

votes

**1**answer

185 views

### Which combinations of normality, separability, and paracompactness do complex manifolds possess?

I am interested in what kinds of non-paracompact complex manifolds may exist and which topological properties they may have.
Is there a non-separable complex manifold? Can a non-separable complex ...

**4**

votes

**1**answer

406 views

### “monotone” homotopy?

This is a question about a concept that I call "monotone homotopy" which arises in a natural way in some topological situations.
Let $X$ be a (bounded) metric space, $Y$ be a topological space and $A\...

**3**

votes

**1**answer

175 views

### Sufficient criteria for $X \subset \mathcal{H}$ to be a Lipschitz (or unif. cont.) retract of $\mathcal{H}$

I am interested in sufficient criteria which ensure that a subset $X$ of a Hilbert space $\mathcal{H}$ is a Lipschitz (or at least uniformly continuous) retract of $\mathcal{H}$.
Under which ...

**3**

votes

**1**answer

497 views

### extending continuous functions from dense subsets to quasicompacts

I am interested under what assumptions one can always extend continuously a function defined on a dense subset; the range of the function is compact but not necessarily Hausdorff.
That is, I am ...

**2**

votes

**1**answer

73 views

### Every $b$-discrete space $X$ with countable injective weight is 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**

vote

**1**answer

72 views

### Non-homeomorphic computable metric spaces whose computable points are computably homeomorphic

This is a follow-up of sorts to an earlier question on mine in that it should be easier to construct a positive example of this, if it exists.
To be clear about definitions, a computable metric space ...

**0**

votes

**1**answer

112 views

### Smallest collection of linear operators satisfying isometry property

Let $\mathscr{A}=\{(A_{1,1},A_{1,2},A_{1,3}),...,(A_{S,1},A_{S,2},A_{S,3})\}$ be a collection of linear operators $A_{n,k}:\mathbb{R}^2 \rightarrow \mathbb{R}^2$. For $u$ and $v\in (\mathbb{R}^2)^3$, ...

**0**

votes

**1**answer

287 views

### Topological properties of complex valued Riemann sum limit curve and a particular integral inequality

I am studying under what conditions the following integral inequality would hold ($a$ real, $a>0$):
$$ \int_{-\infty} ^{\infty} \frac{f(ix)}{a\pm ix}dx\ = 0 \ \ \ \ \Rightarrow \ \ \ \int_{-\...

**0**

votes

**0**answers

113 views

### Minimum rank of certain matrices

Let $\mathscr{M}[n]$ be collection of $n\times n$ matrices with real entries from $\{0,1\}$ such that every row is distinct and every column is distinct.
What is minimum real rank of matrices in $\...

**0**

votes

**0**answers

177 views

### Idempotent ideal in ring of continuous functions

Is there any equivalence conditions under which an ideal $I$ in ring of continuous functions be be an idempotent ideal?

**0**

votes

**0**answers

152 views

### Pro-constructible subset of scheme intersects very dense subsets?

Let $X$ be a scheme, let $D$ be a very dense subset of $X$ and let $Y$ be a pro-constructible subset of $X$. Is it true that $Y \cap D \neq \emptyset$?
If $Y$ is just constructible, this is true.
...

**0**

votes

**0**answers

107 views

### Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space

Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map $...

**0**

votes

**0**answers

147 views

### extension of function in an abstract metric space

my question is the following.(Maybe my title is not quite proper for this question):
Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set $...

**0**

votes

**0**answers

72 views

### Can a “weak” topological space be a Moore space?

Let B be a reflexive and infinite dimensional real Banach space-which could be Hilbert space l^2- and let B be endowed with the weak topology. Although this topology is regular and Hausdorff, it is ...

**0**

votes

**0**answers

192 views

### A noncommutative analogy of the tube lemma

Assume that $A$ and $B$ are two unital commutative Banach algebras. Assume that $\phi \in \mathcal{M} (A)$ is an element of the maximal Ideal space. Define $\alpha: A\hat{\otimes} B \to \mathbb{C}\...

**0**

votes

**0**answers

102 views

### Characterising singular homology among a more general class of cosimplicial spaces

Is there a way to characterise (up to isomorphism) the cosimplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...

**0**

votes

**0**answers

104 views

### minimal (strongly) KC

If P is a topological property, then a space (X, τ) is said to be minimal P (respectively, maximal) if (X,τ) has property P but no topology on X which is strictly smaller (respectively, strictly ...

**0**

votes

**0**answers

80 views

### Extending coverings over dense subsets

Let $X$ be a metric space with $D⊆X$ a dense subset.
If there is a covering for $D$, under which conditions on the covering is it possible to guarantee that the covering also covers $X$?
For a ...

**0**

votes

**0**answers

242 views

### Is $f$ continuous?

The question is also posted here.
The paper is Mizokami : On characterizations of spaces with $G_\delta$-diagonals
See its Theorem 1, also you can see the picture . http://picpaste.com/a-eaiF4d3t....

**0**

votes

**0**answers

141 views

### Only finitely many fundamental groups in $M(n,k,v,D)$?

Let $M(n,k,v,D)$ denote the class of compact manifolds with $Ric \ge \left( {n - 1} \right)k,vol \ge v,diam \le D$.In 1990,M.Anderson proved that "There are only finitely many fundamental groups among ...

**0**

votes

**0**answers

151 views

### measurable function on a locally compact space for a regular measure

A well known classical fact is that a Lebesgue measurable function on Euclidean space is almost everywhere equal to a Baire class 2 function. A relatively modern reference for this fact is van Rooij -...

**0**

votes

**0**answers

191 views

### When is $\{ x \ge 0 | f(x) \le 0\}$ path-connected?

I'm trying to determine the conditions on $f : \mathbb{R}^n_{\ge 0} \to \mathbb{R^n}$ under which $\{ x \ge 0 | f(x) \le 0 \}$ is path-connected. We can assume that $f$ is continuous and concave.
...

**0**

votes

**0**answers

512 views

### Do homotopic non-intersecting simple closed curves separate the surface?

Let $C_1$ and $C_2$ be two simple closed curves on an orientable compact surface $S$, such that:
They are homotopic to each other.
They are set-theoretically disjoint.
Is $S\setminus(C_1 \cup C_2)$ ...

**0**

votes

**0**answers

445 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 (...

**0**

votes

**0**answers

526 views

### Meaning of Regular Neighborhood for Homology Basis Curves in $S_{g,2}$

I have been trying to understand the meaning of the expression "regular neighborhood" in the context described below, but I'm stuck:
We have a collection of curves $c_i$ for $i=1,2,..,n$ embedded in ...

**0**

votes

**0**answers

179 views

### On Birman-Wenzlyfying the B2 spider

Prelude: First of all, let "S matrix" denote "an abstract 4D tensor satisfying
the usual isotypy rules (with no arrows!)". I'm busy trying to classify all possible
S matrices (paper pending) - it's ...

**0**

votes

**0**answers

311 views

### Finding paths in a path connected space

I'm looking for such literature as exists relevant to the following problem.
Problem Given a compact, path-connected region $E$ on the plane and a positive constant $r$. Find (if possible) a path ...

**-1**

votes

**0**answers

70 views

### Is there a standard definition for this topology setup?

There are two complete metric spaces $(A,d_A)$ and $(B,d_B)$, also $B \subset A$, so there is a third induced metric space $(B,d_A)$. There is a continuous and onto function $e:A\to B$. For any $b \in ...