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 (2)

0
votes
0answers
89 views

Show that $f(\bar A) \implies \overline{f(A)}$.

Def) X:a metric space, $Y\subset X$: a subset. A point $x\in X$ is adherent to Y if $B(x;r) \cap Y \neq \emptyset \quad \forall r > 0.$ Def) $\bar Y := \{x\in X \mid x \text{ is adherent to } Y\}$ ...
0
votes
0answers
47 views

Is $(\mathbb{R},+)$ a smooth manifold?

I feel like $(\mathbb{R},+)$ is, but I'm not really sure. How would I know whether or not it is?
0
votes
0answers
43 views

Examples and counterexamples in dimension theory

I am looking for examples of topological spaces that are interesting from a dimension theoretical point of view - in particular I am looking for these three notions of dimension: Little inductive ...
0
votes
0answers
33 views

fundamental group of $\mathbb{C^*}/\{e,a\}$

I'm taking an intro to topology course, and am having trouble with this question. What is the fundamental group of $\mathbb{C^*}/\{e,a\}$, where $e$ is the identity homomorphism and $az=\overline{z}$. ...
0
votes
0answers
33 views

upper hemicontinuity

Let $g: \mathbb R^2_+ \to \mathbb R_+$ and $h: \mathbb R^2_+ \to \mathbb R_+$ continous functions. For every $ t \in \mathbb R_+$, 1) $g(t, \cdot)$ has a unique maximum at $V(t)$ where $V: \mathbb ...
0
votes
0answers
23 views

How does topological dense subgroup induces properties in the larger group?

Let G denote the group of orientation-preserving isometries of the plane; equivalently, the group of affine transformations of the complex field C of the form $z \rightarrow \alpha z + \beta$ ...
0
votes
0answers
56 views

Why is the vertex called non-manifold vertex?

I am working on triangle meshes in one 3D reconstruction project for a while. I know what one manifold vertex looks like and how to detect them. But I hope to understand the definition of non-manifold ...
0
votes
0answers
14 views

Calabi homomorphism of the disk

There is a fact that the homomorphism $Diff_0^{\infty}(\mathbb{D},\partial\mathbb{D},area)\to \mathbb{R}$ is surjective, we can use Calabi homomorphism to prove it, where ...
0
votes
0answers
24 views

What is the classification theorem of simple Lie groups?

I've seen this thrown around a bit, but I can't find what the theorem actually states? Can anyone help?
0
votes
0answers
49 views

Examples of topologies between norm and weak star

Let $X$ be a normed vector space and $X^\ast$ denote its continuous dual. The norm on $X^\ast$ is given by $\|\varphi\|=\sup_{\|x\|=1}|\varphi(x)|$. The weak star topology on $X^\ast$ is the weakest ...
0
votes
0answers
34 views

Question about singular homology

in order to prove that $H_0(X)\simeq \mathbb{F}$, $\mathbb{F}$ is the unitary commutative ring we have to prove that $C_0(X)/B_0(X)\simeq \mathbb{F}$ since we have that $C_0(X)$ is generated by the ...
0
votes
0answers
21 views

Non-meager set of baire property contains perfect subset

I am in need with some help to understanding a proof. Here is the statement and proof. Let B be the collection of all sets of the baire property in R and M be the meager sets in R. STATEMENT : Let ...
0
votes
0answers
15 views

What's an example of a Quotient/Identification Space for this topological space?

I haven't been able to find an example with numbers anywhere on the internet and was hoping someone could help. If I have $X = \{1, 2, 3\}, \tau = \{\emptyset, \{1\}, \{2\}, \{1, 2\}, X\}$ as my ...
0
votes
0answers
96 views

understand proof of compactness in product topology

I am trying to understand the following reasoning. Call $\mathcal{F_\lambda}$ the set of functions $a:\mathbb{N} \to \mathbb{R}$ for which $Na(i) := \sum_{j \in \mathbb{N}} n_{ij} a(j)\leq \lambda ...
0
votes
0answers
15 views

How to use a base to prove something is sequentially compact.

I know this is not very specific but I'm studying for a topology exam and this is one of the things I need to know how to do. I know that part of the process is showing it converges. I was hoping ...
0
votes
0answers
42 views

Existence of bijective function

Does there exist a bijective map $f$ from $\mathbb R^2$ to $\mathbb R^3$ such that $f$ and $f^{-1}$ are both differentiable ? My answer was that since $\mathbb R^2$ and $\mathbb R^3$ are not ...
0
votes
0answers
24 views

History of vectorial bundles in articles or papers?

I'm looking for an article or book that gives a thorough and interesting history of bundles and vectorial bundles in algebraic topology. I'm looking for it for my own learning, please help its ...
0
votes
0answers
31 views

Antipodal map and parallel transport on $S^3$

I'm reading Thurston and Levy's "Three-dimensional geometry and topology" where they have an informal explanation of what life in $S^3$ would look like. For the following detail that I am concerned ...
0
votes
0answers
29 views

A question about open “balls”

I've been recently learning Topology and I'm struggling to visualize open balls. For instance, on $\mathbb{R}^2$ and $\mathbb{R}^3$ given a metric like say $d_\infty(x,y)=\sup\{|x_1-y_1|,|x_1-y_2|\}$ ...
0
votes
0answers
31 views

Hairy ball theorem, projections and L.I. vectors

I'm trying to understand this paper which proves that not every unimodular row is completable by invertible matrices: Why we have these implications: There are two linearly independent vectors at ...
0
votes
0answers
26 views

Some fundamental relations in topology

Are the following relations correct? $\ \{ Normed\, Vector\, Spaces\} \subset \{Topological\, Vector\, Spaces\} \subset \{Uniform \,Spaces\} \subset \{Topological\, Spaces\}$ Then $\ \{Normed\, ...
0
votes
0answers
22 views

An example of a Lindelöf topological space which is not $\sigma$-compact

I am looking for an example of a Lindelöf topological space which is not $\sigma$-compact. I have looked in Counterexamples in Topology, but, if I am not wrong, all the examples there which meet my ...
0
votes
0answers
22 views

Convex in $ \mathbb{R^n}$

Prove that: [A be a convexe part $(A\subseteq \mathbb{R^n})] \implies [\forall x_1,x_2,...x_n\in A ,\forall\alpha_1,\alpha_2,...\alpha_n\ge0 $ $with$ $ \ \alpha_1+\alpha_2+...+\alpha_n=1 ...
0
votes
0answers
15 views

Constructing a smoothly varying basis without singularities

I am trying to construct a smoothly varying and a differentiable basis to map a vector in $\mathbf{B}:\mathbb{R}^3 \to \mathbb{R}^3$. Given a vector field $\mathbf{n}(\mathbf{x})$ where $\mathbf{n} = ...
0
votes
0answers
25 views

Removing non-isolated fixed points

Is it true that if $f$ is a homeomorphism of ${\Bbb R}^n$, then there are other homeomorphisms $g$ of ${\Bbb R}^n$ arbitrarily close to $f$ (in the compact-open topology) such that every fixed point ...
0
votes
0answers
65 views

How I can prove euler characteristic of this complex is zero

$P$ is the poset of all nonempty subsets of $\{ 1, 2, 3, ....,n\}$ under set inclusion. Show that reduced euler characteristic of $\Delta P$ = $ 0$ I tried induction... but failed.
0
votes
0answers
34 views

How do i show that the Riemann sphere and the one-point compactification of $\mathbb{C}$ are homeomorphic?

Let $\mathbb{C}\cup\{\infty\}$ be the one-point compactification of $\mathbb{C}$. Let $S^2$ denote the 2-sphere in $\mathbb{R}^3$. Defne $\zeta:S^2\rightarrow \mathbb{C}\cup\{\infty\}$ as ...
0
votes
0answers
29 views

Contractions and Fixed Points

I'm working on a question in Munkres: If $f$ is a contraction and $X$ is compact, show $f$ has a unique fixed point. Here's my attempt at a solution so far. $f$ is continuous, choose $\epsilon = ...
0
votes
0answers
41 views

A special filter on cartesian product of sets

The following is inspired by this article in nLab (in attempt to simplify it using my notions of funcoids and reloids, which notation is however outside of the scope of this question). Fix a set $U$. ...
0
votes
0answers
42 views

How to deform a curve in specific manner

I am wondering whether we can deform a path in specific ways continuously i mean if there is a closed piece wise $C^1$ smooth path which has to be deformed to another piece wise $C^1$ smooth path. Let ...
0
votes
0answers
27 views

Cluster Points of a Convergence Space

I'm trying to find the characteristic properties (axioms) of cluster points in a convergence space. I've come up with a minimal two: (let $\mathrm{adh}(\mathcal{F})$ be the set of cluster points of ...
0
votes
0answers
21 views

What can we say about the space just by looking at its Borel sets?

What can we say about a compact space $X$ just by looking at the Borel sets of $X$? In general, it seems that not much but maybe it is still not a bad question. For instance, let $X$ be a compact ...
0
votes
0answers
24 views

$(0,1)^\omega$ homeomorohic to $R^\omega$?

Since $(0,1)$ is homeomorphic to $R$ and an infinite product of homeomorphisms is a homeomorphisn?
0
votes
0answers
48 views

topological vector space of measure functions

Let $(X, \mathcal X, \mu )$ be a measure space, and let $ L(X)$ be the space of measurable functions $f: X \to \mathbb C$. Show that the sets $B(f, \epsilon ,r ): = \{ g \in L(X) : \mu( \{ x : | f(x) ...
0
votes
0answers
35 views

A question about rectifiable curves

Does every rectifiable curve that is a subset of the Euclidean plane have zero two-dimensional Lebesgue measure?
0
votes
0answers
35 views

A vector space with topology generated by a family of typologies each makes it a topological vector space is a topological vector space

Let $V$ be a vector space, and let $(\mathcal F_ \alpha ) _{ \alpha \in A}$ be a family of topologies on V, each of which turning $V$ into a topological vector space. Let $\mathcal F$ be the vector ...
0
votes
0answers
42 views

Contractibility of $ S^2 $

In my lecture notes, it said $ S^2 $ was not contractible. If I think about $S^2$ as balls that are shrinking I can see that it is not contractible because the balls of smaller radii area not "in" ...
0
votes
0answers
36 views

Sequence Criterion of Continuity and Divergent sequences

A function $f : (X, d_X) \to (Y, d_Y)$ between metric spaces is continuous iff $$ \lim_{n \to \infty} f(x_n) = f(\lim_{n\to \infty} x_n) $$ for each convergent sequence $(x_n)$ in $(X, d_X)$. So if I ...
0
votes
0answers
42 views

Learning about Calabi–Yau manifold

I want to Create a 3d animation of Calabi–Yau manifold. I tried to learn it but it deals with some very advanced math. What math knowledge have to know before trying to understand the calabi-yau ...
0
votes
0answers
25 views

Is the set, $\{f\in Y: \left\|f\right\|_{Y} \leq C \ \text{and} \ \left\|(|f|)\right\|_{X}\leq C \}$, closed in $(Y, ||\cdot||_{Y})$?

Put, $X= \text{The space of "nice" complex valued functions on } \mathbb R; \text{that is}, f:\mathbb R \to \mathbb C;$ so that $X$ is Banach space with respect to the norm ...
0
votes
0answers
15 views

What is the topology of pointwise convergence of increasing functions?

Given a set $X$ and a topological space $Y$, the topology of pointwise convergence on the set of all functions from $X$ to $Y$ is the product topology on $\Pi Y_x$, since by definition, this is the ...
0
votes
0answers
17 views

Whats the difference between a barrelled space and a locally convex one?

Wikipedia says that a locally convex space is a topological vector space whose topology is generated by translations of balanced, absorbent, convex sets. Whereas a barrelled space is one where: ...
0
votes
0answers
16 views

Projective topology (reference)

Please suggest me a good reference to study Projective topologies. I just want an introductory exposition.
0
votes
0answers
29 views

A dense subset of a finite group

Let $G$ be a finite group with Zariski topology. Suppose $G=A_1\cup A_2\cup\cdots\cup A_n$, where $A_i$, $1\leq i\leq n$, are pairwise disjoint subsets of $G$ and only $A_1$ is dense in $G$, that is, ...
0
votes
0answers
33 views

terminology: accumulation points, limit point, cluster point

In a topological space $X$, what would be the most common terms to describe the following two properties about a point $x\in X$ and a subset $S\subseteq X$. I) For every open set $U$ with $x\in U$, ...
0
votes
0answers
37 views

X is a compact topological space?

Let $X$ be a topological space. Then, when it is said that $K \subset X$ is compact, it is clear to me that every open cover of $K$ contains finite subcover of $K$. But what do we mean when it is ...
0
votes
0answers
23 views

Can the Hjalmar Ekdal topology be defined on uncountable sets?

Can the Hjalmar Ekdal Topology be defined on uncountable sets and how would the various topological properties change from those associated with the set of positive integers? (Example 55 in ...
0
votes
0answers
29 views

What kind of norm is it in the definition of $S^{n-1}$?

Definition (wikipedia) $S^n\triangleq\{x\in\mathbb{R}^{n+1}: ||x||=1\}$ is said to be a 'n-sphere' What norm is it referring to ? I have proven that ...
0
votes
0answers
78 views

Totally bounded uniform spaces vs proximity spaces (need proof)

nLab says "The category of totally bounded uniform spaces and uniformly continuous functions is equivalent to the category of proximity spaces and proximally continuous functions". How to prove this? ...
0
votes
0answers
25 views

Cauchy filters defined for proximity spaces?

I define in my draft article Cauchy filters $\mathcal{X}$ on a uniform space $\nu$ by the formula: $$\mathcal{X}\ne\bot \wedge \mathcal{X}\times^{\mathsf{RLD}}\mathcal{X}\sqsubseteq\nu.$$ ...