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)

0
votes
0answers
22 views

lifting (not just homotopies!) theorems for fibrations?

Is there an analogue of the lifting theorem for covering spaces in the situation of fibrations? In particular, given a fibration $f : E\rightarrow S$ and an automorphism $\sigma\in Aut(S)$, can you ...
0
votes
0answers
23 views

Peano curve and Peano's original paper

I have a question regarding the original paper of Peano (here is a link), where he defined his curve in terms of ternary expansions and a mirroring operator. In short, he describes there a continuous ...
0
votes
0answers
31 views

Proof that set of functions with derivative zero at a given point is meager in space of strictly increasing twice differentiable functions.

Let $X = \{f: [0,1] \to \mathbb{R} \; | \; f\in C^2[0,1], f \textrm{ strictly increasing} \}$. I equip $X$ with the topology of uniform convergence. Define the set $A$ as: $$A =\{ f \in X \; | \; ...
0
votes
0answers
39 views

Generalization of a class of sets

In topological space, we start with open set, which serves as fundamental set. We know that union of finite disjoint open sets is the smallest set amongst any kind of unions of open sets, so we have a ...
0
votes
0answers
58 views

sequential continuity and countinuity

When we have two topological spaces, $\left(X, \tau_X\right)$ and $(Y, \tau_Y)$ it is easy to check that for $f: X \rightarrow Y$ continuity implies sequential continuity. I'm wondering what do we ...
0
votes
0answers
49 views

help in proving equivelant statements of $f=\sum\limits_{i=1}^\infty \langle f,\phi_i \rangle \phi_i \space \forall f\in H$

Let $H$ is a hilbert space. $\{\phi_i\}_{i=1}^\infty$ is an orthonormal set A set $\{\phi_i\}_{i=1}^\infty$ is complete in $H$ if any of the following statements hold: $f=\sum\limits_{i=1}^\infty ...
0
votes
0answers
19 views

$A\subseteq B(X, Y)$ compact if and only if closed and $Ax$ is conditionally compact

This comes from Exercise 2 of Chapter VI in Dunford & Schwartz. I am trying to prove the following statement: A set $A\subseteq \mathscr{B}(X, Y)$ is compact in the strong operator topology if ...
0
votes
0answers
50 views

$F \subset X= \prod_{\alpha}x_\alpha \ \alpha \in I$ is closed if and only if $F$ is the finite intersection of…

I'd just like to see if the following proof is reasonable - and if not, what went wrong. Thx for taking a look! $F \subset X= \prod_{\alpha}x_\alpha \ \alpha \in I$ is closed if and only if $F$ is ...
0
votes
0answers
51 views

Gromov compactness theorem

Reference: this book, page 493. For a compact metric space $X$ define $\text{Cov}(X,\epsilon)= \min \{n \, : \, X \text{ is covered by $n$ closed } \epsilon\text{-balls} \}$ and ...
0
votes
0answers
22 views

3-manifold with boundary and corners

In the literature I often come across the sentence "3-manifold with boundary and corners" but I am not sure what does that mean? To be more specific: - What types of boundary can a generic 3-manifold ...
0
votes
0answers
15 views

The index number of an isolated critical point = 1+(e-h)/2

I know that the number of elliptic (e) and the number of hyperbolic should be even. But i find a difficulty in coming up with a proof for this
0
votes
0answers
24 views

Torus linking and natural geometric on the torus

I have the three-dimensional manifold $$X=\lbrace(x,y,z)\mid x\neq y \neq z \neq x \rbrace\subseteq S^1\times S^1 \times S^1 $$ which we can see it as the configuration space of 3 points on a ...
0
votes
0answers
16 views

Closure of a set with specified distance condition

Salam. I've presented the question and my thoughts on it. The question states: Let $S$ be a subset of $\Bbb R$ and $a \in \Bbb R$. Prove that $a \in \overline{S}$ if and only if for each positive ...
0
votes
0answers
36 views

Proving version of Stone Weierstrass for locally compact space

Let $X$ be a locally compact Hausdorff (LCH) space. Suppose that $\mathcal{A}$ is a closed algebra of $C_0(X)$ (the continuous real-valued functions on $X$ with compact support). Suppose in addition ...
0
votes
0answers
55 views

Prove the Bolzano-Weierstrass theorem using Cauchy sequence property

Prove the Bolzano-Weierstrass theorem using the Cauchy sequence property. If $A$ is closed and bounded then $A$ has a convergent subsequence. Cauchy sequence definition: A sequence ...
0
votes
0answers
24 views

Relation between closed sets and accumulation points

Can a closed set have no accumulation points? My textbook defines the closure of a set $A$, as: $A\cup$ Acc. Points.
0
votes
0answers
23 views

What tools in algebraic topology help me capture the connectivity structure of a weighted graph as a REAL number?

All - This is a follow-up to a previous question about cohomology. I am researching a problem and, as with so much problem-solving, this has led me into parts of math well beyond where I went in ...
0
votes
0answers
34 views

Definition of Diffeomorphism for Arbitrary Subsets of Euclidean Spaces

In pg 1 of Chapter 1 of Milnor's Topology From the Differentiable Viewpoint, it is defined that Definition. Let $f:X\to Y$ be a function from a subset $X$ of $\mathbf R^k$ to a subset $Y$ of ...
0
votes
0answers
47 views

A basis for the profinite topology

Let $G$ be a group, and let $$C =\{H\leq G :\ [G:H]<\infty \}$$ $$B_R =\{Hx\leq G :\ H\in C ,x\in G\}$$ $$B_N =\{Hx\leq G :\ H\in C ,H\ is\ normal\ in\ G,x\in G\}$$ I want to show that $B_R$ ...
0
votes
0answers
29 views

Euler Characteristic of Side identification

Let $S$ be a surface obtained by identifying the sides of a regular hexagon in pairs. I want to show $\chi(S) > -1 $. I can see how we can obtain surfaces with $\chi(S) = 0,1,2$ but I think I'm ...
0
votes
0answers
44 views

coloring theorem for topological partitions

Let $(X,\tau)$ be a topological space. DEFINITIONS: Define a topological partition of $X$ into connected sets to be a collection of pairwise disjoint open connected sets $\{U_i\}_{i\in I}$ such that ...
0
votes
0answers
23 views

In the category of uniform spaces, is the completion of a quotient map also a quotient map?

Let $X$ and $Y$ be two Hausdorff uniform spaces. A surjective uniformly continuous map $f: X \to Y$ is a quotient map if for every map $g$ from $Y$ to a uniform space $Z$ such that $g \circ f$ is ...
0
votes
0answers
58 views

The simple meaning behind covering map & lifting

When it comes to math concepts, my sneaky feeling is that they were all started up as simple ideas, but got more and more convoluted as mathematicians tried harder and harder in making them more ...
0
votes
0answers
42 views

Homeomorphism on a Preordered Set

Prove that if $h$ is a homeomorphism from $(X,\mathscr{S})$ to $(Y,\mathscr{T})$, then $\forall a, b \in X \left( a \trianglelefteq_{\mathscr{S}} b \iff h(a) \trianglelefteq_{\mathscr{T}} h(b) ...
0
votes
0answers
44 views

CW-complex definition

I've the following definition a (finite) CW-complex of dimension $N$ is a topological space $X$ defined in the following way: $X^0$ is a discrete set of points $\forall 0<n\le N$, $X^n$ is ...
0
votes
0answers
41 views

Proof of Banach's homeomorphism theorem without the contraction map principle.

Let $E$ a Banach's space and $X\subset E$ open. The Banach's homeomorphism theorem tells us that if a function $F:X\to E$ is a contraction on $X$ then $(I+F):X\to E$ is a homeomorphism of $X$ onto ...
0
votes
0answers
28 views

Size of largest fiber

Suppose I have a map $f\colon X\to Y$ with finite fibers (where $X,Y$ are simply topological spaces for now, although maybe it's better to think of schemes). Is there a name for the quantity ...
0
votes
0answers
38 views

Uniqueness for a covering map lift: is locally connected necessary?

So I just got through proving the following theorem: If $p:C\to X$ is a covering map and $Y$ is a [xxx] space, then given $y_0\in Y$, $c_0\in C$, $f:Y\to X$ such that $f(y_0)=p(c_0)$ there exists ...
0
votes
0answers
20 views

Identifying assumptions in a proof of a limit

I'm following a proof which starts with following inequalities: $$ dvP_{i}(v+dv) \geq {S_{i}(v+dv) - S_{i}(v)} \geq dvP_{i}(v)$$ where $v\in S$ and $v+dv \in S$ Diving by $dv$ and as $lim_{x \to 0}$ ...
0
votes
0answers
31 views

So i read that kannan maps are caristi maps, how do I prove it?

Let $(X,d)$ be a complete metric space and $f:X \rightarrow X$ be a function such that $d(f(x),f(y)) \leq k(d(x,f(x))+d(y,f(y)))$ for $k \in[0,\frac{1}{2})$ I have tried to prove the following ...
0
votes
0answers
32 views

Colorings of Topological Partitions (color-boundedness)

Let $(X,\tau)$ be a topological space. DEFINITIONS: Define a topological partition of $X$ into connected sets to be a collection of pairwise disjoint open connected sets $\{U_i\}_{i\in I}$ such that ...
0
votes
0answers
43 views

Subbase for a topology defined by a family of closed sets?

Let $X$ be a set and $\mathcal{T}$ the smallest/weakest/coarsest topology containing some family $\left\lbrace C_i\right\rbrace_{i\in I} $ of sets declared to be closed. This is well-defined since ...
0
votes
0answers
21 views

Seminorms: Continuity

Problem Given a topological vector space $V$. Consider a seminorm with cylinder: $$\mu:V\to\mathbb{R}_+:\quad B_\mu:=\{\mu<1\}\subseteq U\subseteq\{\mu\leq1\}=:\overline{B}_\mu$$ Then one has as ...
0
votes
0answers
20 views

Connected Plane of integers

I am curious about the plane of integers, that is $(x,y)$ where $x,y\in\mathbb{Z}$ as a subset of $\mathbb{R}^2$. Is there any way to prove it is not connected or connected? How would one go about it? ...
0
votes
0answers
26 views

Question about a cycle when doing topological sorting

First off, this is a homework question, but I'm just a bit confused on some of the smaller details of doing a topological sort. The homework question can be seen here (it shows the graph). It says: ...
0
votes
0answers
26 views

Translation of GENUS to Portuguese

Does somebody know a translation (to portuguese) for "genus" in topology? Theorem: A nonsigular projective curve in $\mathbb{P}_2$ is topologically a sphere with $g$ handles. Definition: This number ...
0
votes
0answers
23 views

Intersection of Decreasing closed sets

If $\{E_n\}_{n \in N}$ is a sequence of closed nonempty and bounded sets in a complete metric space $(X,d)$, if $E_n \supset E_{n+1}$ and if $$\lim_{n \to \infty} \operatorname{diam} E_n = 0,$$ then ...
0
votes
0answers
26 views

an open subspace of locally compact is dense

Let $X$ be locally compact Hausdorff. Then a subspace $A$ of $X$ is dense and locally compact iff $A$ is open. I can prove the necessary condition. But for the sufficient condition, I can not get ...
0
votes
0answers
25 views

Definition of a topology on a set $X$: may $I$ contain uncountable many elements or is it restricted to be finite or countable?

I've the following definition of a topology $\mathcal I$ on a set $X$: (T1) $U_a \in \mathcal I, a \in I \Rightarrow \cup_{a \in I} U_a \in \mathcal I$ (T2) $U_1, U_2 \in \mathcal I \Rightarrow U_1 ...
0
votes
0answers
23 views

Simple question about of the formulas concerning the forcing relation $U \Vdash A$

If $A$ is comeager in $U$, we say that $U$ forces $A$, in symbols $$U \Vdash A$$ A weak basis for a topological space $X$ is a collection of nonempty open sets such that every nonempty open set ...
0
votes
0answers
22 views

How two disjoint solid 2-tori linked?

I have 3-manifold which is a union of two disjoint solid 2-tori, How I can decide if they are meet or no? Also if they are meet how I can know the way they are linked?? Thanks in advance
0
votes
0answers
72 views

Is any closed and bounded subset of a reflexive Banach space compact in the weak topology?

It seems to me that Alaoglu's theorem implies that any closed and bounded subset of a reflexive Banach space is compact in the weak topology. Is convexity of the set also needed?
0
votes
0answers
22 views

Inclusions in typologies induced by euclidean and square metric

In Theorem 20.3, page 123 Munkres Topology(second edition) the following is mentioned. Inequality $\rho (x,y) \leq d(x,y) \leq n^{1/2}\rho (x,y)$ implies that $B_d(x,\epsilon)\subset ...
0
votes
0answers
20 views

Perfect subset of the group of characters of an additive dense subgroup of $\mathbb(R)$

Suppose that $\Gamma$ is a dense subgroup of the group of real numbers $\mathbb{R}$ and let $G$ be the group of characters of $\Gamma$. The problem is to show that for any $a\in\Gamma$ the set ...
0
votes
0answers
17 views

Hemicontinuity of correspondence

I have a correspondece $f$ of $\mathbb{R}$ in $\mathbb{R}$ that gives me $\textrm{sin}(1/x)$ if $x \neq 0$ and $[-1,1]$ if $x=0$. I have to test whether this function is continuous or not, using ...
0
votes
0answers
26 views

When the preimage of the family of all open neighborhoods of a point is cofinal

Let $f \colon X \to Y$ be a continuous map of topological spaces. Denote by $O_y(Y)$ the family of all open neighborhoods of a point $y \in Y$. Define $$ f^{-1}(O_y(Y)) = \bigl\{ f^{-1}(U) \colon U ...
0
votes
0answers
35 views

Boundary and Interior Points?

For each of the following sets $S$, find the boundary, $bd S$, and the interior, $intS$. a. $S = [0,2] \cup (2,4)$ b. $S = [0,2] \cap (2,4)$ Workings: a. $S = [0,2] \cup (2,4)$ $bd S = [0,2] ...
0
votes
0answers
29 views

Boundary of a set in topology

I understand the boundary to mean all the points that are in the set AND not in the set. For example in $\mathbb{R}^2$. $S=\{(x,y)\; | \;(x^2)-x\leq y \leq 0\}$. This is a parabola that opens upward. ...
0
votes
0answers
37 views

Showing deleted integers topology not locally path connected

Why is the deleted integer topology not locally path connected? It is listed in the book "counterexamples in topology" However, I cannot seem to find a point which is not locally path connected since ...
0
votes
0answers
51 views

Manifold and the topology of $\mathbb{R}^{n}$

A manifold $M$ is defined in particular as being locally homeomorphic to $\mathbb{R}^{n}$. Homeomorphisms can be defined in terms of how they map open sets, namely an homemorphism $f$ and its inverse ...