Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

learn more… | top users | synonyms (3)

2
votes
0answers
101 views

Discontinuous function on Q

let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ ...
2
votes
0answers
70 views

When is a function space a Fréchet space?

Let $Q$ be a space of indices, and let $(V, |\cdot|)$ be a Banach space of values. Define the function space $X = C(Q,V)$, and equip it with the topology generated by seminorms $\|x\|_D := \sup_{d \in ...
2
votes
0answers
56 views

Morse-Smale Complex, boundary on the number of segments by the number of critical points.

I am looking for a known upper bound on the number of monotone regions of a Morse function by the number of its critical points in the interior of the manifold and on its boundary. Here I try to ...
2
votes
0answers
44 views

Is an ideal generated by a compact subset finitely generated?

Let $R$ be a commutative topological ring and let $K$ be a compact subset of $R$. Denote by $I$ the ideal generated by $R$. Then is it true (or under what assumptions on $R$ (besides Noethernity)) is ...
2
votes
0answers
222 views

A domain on a sphere is simply connected if and only if its complement is connected

I think the statement that a domain (open connected set) in a sphere is simply connected if and only if its complement is connected is a standard result. But how can one prove it? Is it possible to ...
2
votes
0answers
119 views

Locally / Strongly connected product spaces

I have to give necessary and sufficient conditions s.t. $ X=\prod_i X_i$ is locally / strongly connected, where $(X_i,\tau_i)$ are non-empty top. spaces. First locally: Assume all $X_i$ are locally ...
2
votes
0answers
112 views

Set of limit points of S is closed in a metric space X

A point $x \in X$ is a limit point of a subset S of X, if every ball $B(x;\varepsilon)$ contains infinitely many points of S. Show that x is a limit point of S iff there is a sequence {$x_{j}$} ...
2
votes
0answers
78 views

Exploiting the compactness of the unit circle to prove the following proposition.

I am trying to prove that a locally convex topological vector space is equivalent to a semi-normed topological vector space. I have worked through the proof but I am unsure because of the following ...
2
votes
0answers
69 views
2
votes
0answers
160 views

Point set topology from an algebraic perspective?

I got this idea of viewing a topology as an operation on a ring of sets. Let $\mathcal R = (\mathcal P(X), \cap, \triangle)$ be a ring of sets. ($\triangle$ is the symmetric difference operation and ...
2
votes
0answers
31 views

A question on the classical Mrowka space

Definition: A space $X$ is $\Delta$-normal if for every $A \subset X^2 \setminus \Delta_X$ closed in $X^2$ there exist disjoint open $U$ and $V$ in $X^2$ such that $A \subset U$ and $\Delta_X \subset ...
2
votes
0answers
149 views

hyperspace of a complete uniform space need not be complete

I want to know the counter example for: Hyperspace of an arbitrary complete uniform space need not be complete. The hyperspace of a uniform space $(X,\mathscr D)$ is obtained by forming the set ...
2
votes
0answers
47 views

What are default topologies on $R^∞$ and $R^ω$?

To extend the original question Difference between $R^\infty$ and $R^\omega$: What are default topologies on $R^∞$ and $R^ω$? I would think that we take product topology on $R^ω$ and limit topology ...
2
votes
0answers
124 views

Existence of points in closed and bounded convex sets that cannot be expressed as convex combination of other elements of the set.

I have an intuition about convex, closed, bounded sets but I can't really find a way to prove whether it's right or wrong. Let $\Sigma$ be a convex set, that means, that given any $A,B \in \Sigma$, ...
2
votes
0answers
41 views

Knowing the existence of a fixed point set from an induced fundamental group automorphism

Let $L$ be a link in $S^{3} $ and $f_{ \phi } : \pi_{1} (S^{3} \backslash L ) \rightarrow \pi_{1} (S^{3} \backslash L )$ be induced from a periodic map $\phi $ of $S^{3} $, restricted to the ...
2
votes
0answers
72 views

Interpreting Finite Topologies

Taking the quotes Intuitively, an open set provides a method to distinguish two points. For example, if about one point in a topological space there exists an open set not containing another ...
2
votes
0answers
147 views

Why second countable for definition of manifold?

What is the motivation to define a manifold to be second countable? What kind of pedagogical issues does this avoid?
2
votes
0answers
81 views

How do you specify a link to a blind combinatorialist?

Regular projections of links look like graphs in the plane. So I'm wondering if it would be possible to specify a link up to isotopy with purely combinatorial data about this graph. If so, what kind ...
2
votes
0answers
136 views

Continuity of functional

This is a follow up question to this previous one. Let $X\subset\mathbb{R}_+$, where $X$ is countable and we are considering the space $\mathbb{R}^X$ of functions $f:X\to\mathbb{R}$ with the topology ...
2
votes
0answers
48 views

What is the name for the topology where every point is in the boundary of an open set?

Is there a name for topological spaces in which every point is in the boundary of an open set?
2
votes
0answers
46 views

The characterization of asymptotic dimension

Let X be a metric space. The following conditions are equivalent (a)asdimX = n (b)n is the smallest integer such that for every R > 0 there exists n + 1 families Ui i=0,1,2,...,n, and S > 0 such ...
2
votes
0answers
109 views

Is the category of topological spaces coregular?

Everything is in the title. The category Top of topological spaces and continuous mappings is not regular, but is it coregular ? Furthermore, Top isn't cartesian closed, but does it satisfy the dual ...
2
votes
0answers
109 views

Subgroup Separability translated in Profinite Topology

The normal definition of subgroup separability is: A group $G$ is said to be subgroup separable if for every finitely generated subgroup $H\leq G$ and $g\in G\setminus H$ there exists a subgroup of ...
2
votes
0answers
76 views

Product varieties with the constructible topology

Let $k$ be an algebraically closed field and let $X\subseteq k^n$, $Y\subseteq k^m$ be two affine algebraic varieties. It is not difficult to find examples where the Zariski topology on the product ...
2
votes
0answers
80 views

From Jordan's Curve Theorem to Jordan-Schoenfliess theorem

I am trying to learn and understand proofs of classical theorems and successfully mastered a proof of JCT. (It was the well-known proof that uses Tietze Extension and Brouwer's fixed point theorem). ...
2
votes
0answers
162 views

Orientability as a topological property

Can one prove that orientability(of a manifold)is a topological property without using algebraic topology? That is, using a combination of general topology,linear algebra,and topological groups(such ...
2
votes
0answers
266 views

Construction of Lakes of Wada

At each step of the construction of Lakes of Wada we extend a lake (an open set in the open unit square) so that no point of the land (the complement of all the lakes) is farther than a given small ...
2
votes
0answers
54 views

Quotient of complete linearly topologized ring

The quotient of a complete metrizable group by a closed normal subgroup is always complete, but there are examples to show this need not be true for non-metrizable groups. Here complete means every ...
2
votes
0answers
81 views

About the universal bundle $EG\rightarrow BG$

For a topological group $G$, we define $EG$ to be the infinite join of $G$, and $B$ to be the quotient of $EG$ by the left action of $G$. Explicitly $EG$ can be expressed, as a set, as ...
2
votes
0answers
54 views

Inverses of two argument functions with respect to one argument

Consider a function $f : A \times B \to C$ and two inverses, each with respect to one argument; i.e. $g$ and $h$ defined such that $f(x,y)=z \iff g(y,z)=x \iff h(z,x)=y$. A simple example is addition: ...
2
votes
0answers
44 views

Is there a name for a set where any two elements are separated by a given distance?

I am curious if there is a name for such a set. Let $(M,d)$ be a metric space and $S$ a subset of $M$ for which there is some positive number $\delta$ such that for any two distinct elements in $S$, ...
2
votes
0answers
230 views

An exercise from the handbook of set-theoretic topology

This is an exercise from the handbook of set-theoretic topology (Exercise 13.3): Assume $\mathfrak b=\mathfrak c$. Construct a first countable separable zero-dimensional locally compact ...
2
votes
0answers
79 views

derivatives using epsilon-delta argument

I have this question below and am not sure if my approach is correct. Can anyone please advise me? Thanks. Question: Let $f:\mathbb{R}^n \to \mathbb{R}^m$ and suppose there is a positive constant $K$ ...
2
votes
0answers
56 views

Finding a sufficient condition for a set to be finitely decomposable into open sets..

Let us call a set in a topological space finitely decomposable set (FDS) iff it can be rewritten using the standard set operations $\cup$ and $\sim$ and only finitely many open sets. I'm looking for ...
2
votes
0answers
123 views

Pushing a map off a disk

Let us assume that we have covered $\mathbb{R}^n$ with the open sets $V = 2 \cdot int(D^n)$ (the standard unit disk, and 2 means multiply the size by 2) and the family $\mathcal{U}$ of open disks W ...
2
votes
0answers
33 views

Let $(X,\tau)$ be a $T_B$-space…

A topological space is called $T_B$ if every compact subset is closed. (I):$Let (X,\tau)$ be a $T_B$-space which is not countably compact, $\{x_n :n \in \omega\}$ a set without accumulation points, ...
2
votes
0answers
76 views

What is the use of locally connected spaces?

One of the main properties of locally connected spaces is that their connected components are clopen and thus, they are homeomorphic to the colimit of their connected components. This is good to ...
2
votes
0answers
86 views

Compact Hausdorff implies product of quotient map is a quotient map?

Let $X$ be compact Hausdorff and let $q : X \to Y$ be a quotient map. Is it true that $f : X \times X \to Y \times Y$ with $(x_1, x_2) \mapsto (q(x_1), q(x_2))$ is a quotient map?
2
votes
0answers
312 views

“is topologically mixing” vs. “is topologically transitive” in the defition of chaos

Wikipedia gives essentially "is topologically mixing and has dense periodic periodic orbits" as the definition of chaos, and this paper shows that its (the paper's) definition of chaos is equivalent ...
2
votes
0answers
62 views

Isotopy between two open disks on a surface

So I have a (compact) surface $\Sigma$ and two open disks on the surface call them $A$ and $A'$ such that the intersection contains a simple curve $P$. What I want to do is construct an isotopy ...
2
votes
0answers
66 views

Visually apealing holologous transformation of a given contour

There is this problem which roughly says: You want to put a framed picture onto the wall with a cord to the picture frame. The cord is a single one, and both ends are attached to the frame. ...
2
votes
0answers
72 views

Direct construction of a lifting

Let $f : \mathbb R^2 \to S^1$ be a continuous map such that $f(0,0) = 1$. Using standard theorems about the existence of liftings (e.g., Proposition 1.33 of Chapter 1 in Hatcher's book Algebraic ...
2
votes
0answers
146 views

Is my general approach to proofs acceptable? A general topology example.

Proving: $A$ is closed iff $A = \bar{A}$. "To the right": If $A$ is closed, $ A = \bar A$ If $A$ is closed this means that it contains all of its own accumulation points. And we would find that its ...
2
votes
0answers
87 views

Counter-example about paracompactness

I am trying to find a counter-example related to the definition of paracompactness, but it seems that it is not very easy. Here is the problem. Give an example to show that if $X$ is paracompact, ...
2
votes
0answers
207 views

Intuition behind continuity in topological spaces

I was approaching the following problem: "Let $f \colon X \to Y$ be continuous. Is it true that if $x$ is a limit point of $A \subset X$ then $f(x)$ is a limit point of $f(A)$?" The answer is that ...
2
votes
0answers
107 views

Hartshorne II Prop 2.6

Prop 2.6 constructed a continuous map $X$ to $t(X)$, I cannot verify that it is a homeomorphism. I try to show any open set $U$ is mapped to $t(X)\setminus t(X\setminus U)$. To show it is surjective, ...
2
votes
0answers
49 views

Looking for articles on postcritically finite rational maps in Russian or French

I'm looking for articles on postcritically finite rational maps. I found a few articles in English, but I can't find any in Russian or French.
2
votes
0answers
84 views

Is this case possible (hedgehog metric, colinearity)

My topology class was asked to prove that the hedgehog metric was indeed a metric (the details are irrelevant for my question). This does not concern the proof itself, but rather the structure of the ...
2
votes
0answers
54 views

Not 1-dimensional homological equivalent of the circle

The questions origins from this problem and my incorrect answer to it. I'm trying to correct it, but it turned out that the topological space - that I need to do it straightforward - has very specific ...
2
votes
0answers
70 views

What is the relation between singular point for a function and the one in a vector field?

What is the difference between sigular point for a function and the one in a vector field? Is the derivative or divergence at the singular point must be infinity? By the way, what is the relation ...