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; ...

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164 views

Countable Product of discrete spaces

Let $X$ be a countable discrete topological space. Consider $X^{\mathbb{N}}$ endowed with the product topology. How do you prove that $X^{\mathbb{N}}$ is homeomorphic to the sub-space of all ...
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194 views

Can we find a nice definition of Congruence in Topology?

According to my knowledge, quotient structure is a original structure divided by a congruence. However, quotient topology space is defined this way. Quotient_topology In this way, $\sim$ is only said ...
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95 views

Berkovich analytification of Robinson fields

Let $\rho$ be an infinitesimal and let $^\rho \mathbb{R}$ be a (non-archimedean) Robinson valued field. Is there anything known about the topological structure of $\mathbb{A}^{1,an}_{^\rho ...
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734 views

Are the continuous functions pointwise dense in the bounded measurable functions

Suppose we have a compact set $K$. I know that the space $C(K,\mathbb{C})$ of continuous functions is complete with respect to the norm $\|f\| = \sup_{x\in K} |f(x)|$. Let $L^{\infty}$ be the space of ...
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224 views

Applications of monads in general topology?

What are applications of monads in general topology? For example, for GT is important the notion of products, products are adjoints, so adjoints may be important for GT, but what's about monads?
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76 views

JSJ-decompositions of groups and 3-manifolds: a reference request

I am, for whatever reason, interested in learning about the JSJ-decomposition of groups. Having asked around a bit, it was suggested I first learn about what is happening in the manifolds and then ...
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502 views

Showing that metric induces single unique topology on a finite set

I am trying to prove, that given a metric on a finite set it induces exactly one topology. I have an idea which might lead to a proof, but am not sure: For a finite set X with a given metric d we can ...
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89 views

Total sets in $R$ compared with total sets in $[a,b]$

A total set in a NLS is one whose linear span is dense in the set. e.g. $A = \{1,x, x^2,...\}$ is total in $(C[a,b],\Vert\cdot\Vert_{\infty})$ I find it easier to talk about total sets than dense ...
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104 views

Image of Thom Class under Sequence of Maps?

So I've been trying to do problems in Milnor & Stasheff's Characteristic Classes as a quick review, not having done anything with them in a while. However, I'm stuck on some parts in attempting ...
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673 views

Extend by Continuity

This is a very short question, I hope it is not too broad, if so I shall try and make it more specific. I would like to start as it stands below, though, because it really points down the essence of ...
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344 views

The topology of distributions

I have been wondering about the following concerning the spaces $\mathcal D$ of test functions (say on $\mathbf R$). It is my understanding that the topology on this space is inductive limit ...
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519 views

Topology of wedge products

I have a question about the quotient topology induced on the wedge sum $S^{\,2} \vee S^1$, (where $S^n$ denotes the unit sphere in $\mathbb{R}^n$). In this topological space, the subsets $S^1$ and ...
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104 views

A Heegaard splitting of $S^2\times S^1 \# S^2\times S^1$.

For a Heegaard splitting of $S^2 \times S^1$, we can take two copies of genus 1 handlebodies and glue boundaries with the identity map. I want to generalize this a little bit. In the case of ...
2
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252 views

Relations and differences between outer/inner limit and Kuratowski limsup/liminf

Let $X$ be a topological space. I am asking about the relations and differences between the following two different types of $\limsup$ and $\liminf$ of $A_n ⊆ X, n ∈ \mathbb{N}$, a sequence of ...
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351 views

Vector calculus- vector field and path

Let $U\subset \mathbb{R}^2$ be open, and $F:U\to \mathbb{R}^2$ a $C^1$-vector field. Assume that: $$\frac{\partial{F_1}}{\partial{x_2}}(x)=\frac{\partial{F_2}}{\partial{x_1}}(x)\quad\forall x\in U$$ ...
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90 views

factor of covering map is a covering map?

A paper I'm trying to understand uses the following lemma: Let $p: U \to U_0$ be a topological covering map. Suppose that we can write $p =\pi \circ f$, where $f:U \to Y$ is an open surjective map, ...
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90 views

topology exercise. compactness circle projective space.

Is the circle compact in $\mathbb{P}_{2}(\mathbb{C})$? Here what I did: I considered the circle in $\mathbb{C}^2$ is $\{(x,y)\in\mathbb{C}^2|x^2+y^2=1\}$. The projective closure in ...
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164 views

Trying to prove a intuitively “obvious” fact.

I'm trying to prove that all continuous maps of pairs $f:([-1,1], \{-1,1\})\to (\{-1,1\},\{-1,1\})$ are constant, and I've almost got a working argument, but it reduces down to the following ...
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80 views

Does it imply a lift?

Let $p:S^1\times S^3\rightarrow S^1\times S^3$ be a covering map with $p(z,y)=(z^3,y)$ and $z\in S^1\subset\mathbb{C}$ and $h:\mathbb{R}P^4\rightarrow S^1\times S^3$. Is there a lift ...
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198 views

Why a spiral is the deformation retract of a plane?

As the title says, why a spiral is the deformation retract of a plane?.
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55 views

Is the functor associating a bundle with a structure group to a principal bundle faithful?

Consider a (Cartan) principal G-bundle $\xi: X \to B$, and a left $G$-space $F$. We construct the bundle $\xi[F]: X_F \to B$ associated with $\xi$ with a fiber $F$ as usual. Now for each morphism $(u, ...
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321 views

Do we need net refinements not induced by preorder morphisms?

From Engelking's book on general topology (slightly rephrased): Definition: We say that the net $S': \Sigma' \to X$ is finer than the net $S: \Sigma \to X$ if 1. there exists a function $f: ...
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47 views

Set of 3D surfaces

How might one show that the set of connected 3D surfaces with infinite genus (up to homeomorphism) is countably infinite? I am guessing that we could either use proof by contradiction or come up ...
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263 views

Point-set topology and set theory

In the standard second-order, but single-sorted setting of point-set topology one has a base set $X$ and the property of being open on its powerset $P$ obeying the usual axioms. Proofs in point-set ...
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250 views

Generalized Jordan curve theorem (and a related MAIN QUESTION)

Preliminaries A Jordan map is a continuous map $f: [0,1] \rightarrow \mathbb{R}^2$ such that $f(0) = f(1)$ the restriction of $f\ $ to $[0,1)$ is injective A Jordan curve is a subset $\gamma$ of ...
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416 views

Properties of the universal cover of CW-complexes

Let $Y$ be a CW-complex and $X$ its universal cover. Could you give me a proof (or a referece) for the following fact: $X$ is contractible $\Leftrightarrow$ $H_i(X)=0$ $\forall i\geq2$ ...
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126 views

Homeomorphism groups as topological groups

As is well known, the homeomorphism group of a compact Hausdorff space is a topological group. The same is true for locally compact locally connected Hausdorff spaces, but it is false in general. Now ...
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361 views

Learning analysis through topology

One of my supervisors once mentioned that when he was learning analysis he learnt it backwards. He learnt topology first and then saw analysis after, instead of the usual approach of doing everything ...
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270 views

Direct proof to show that a set is closed

I was trying to prove something, and I did it, but what I used is too exaggerated. The problem is: Let K be the cantor set, prove that the sets $$ \eqalign{ & \left\{ {\left| {x - y} ...
2
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260 views

uniform distribution on unit ball

If $S$ is a set of an countably infinite number of points uniformly distributed throughout the unit ball in $\mathbb R^n$, is there for every point $p$ in the ball and every real number $e>0$, a ...
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131 views

Property (T) for groups vs top

I have encountered two properties in different areas of math. One is the property (T) of groups and the other is the property (T) of topologies. What is the connection between these two ? Thank you. ...
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107 views

Contractions and Map Extensions

I'm going through Spanier and got stuck on the following problem: Show that a space $Y$ is contractible if and only if given a pair $(X,A)$ having the homotopy extension property with respect to $Y$, ...
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0answers
494 views

homeomorphisms mapping interiors to interiors and boundaries to boundaries

Why do homeomorphisms map interiors to interiors and boundaries to boundaries? I cannot find a good proof for it that does not involve algebraic topology. I only need it for spaces in ...
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357 views

How do you prove a CW complex is locally path connected

I think this is done inductively on the skeletons but I can't work out the details.
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266 views

fundamental group of closed surfaces as CW complexes

Let $T$ denote the $2$-torus, $P$ the projective plane, and $nT$/$nP$ the connected sum of $n$ tori/$n$ projective planes respectively. 1) how can I prove, that $nT$ and $nP$ are homeomorphic to ...
2
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146 views

One special set on [0,1]

Let $X = [0,1]$. Define $f:X\to\mathbb{R}_{\geq 0}$ to be Lipschitz continuous on $X$. Put $$Y\subset X:\int\limits_Y f(x)\,dx = 0$$ What can we say then about $A = X\setminus Y$? It is not defined ...
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81 views

Questions about boundary faces of simplices and triangulations

Let $S$ be a simplex in $\mathbf R^n$ and let $\{S_i\}$ be a triangulation of $S$. The boundary of $S$ is defined as the union of the boundary faces of $S$. Is this union equal to the topological ...
2
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147 views

How to describe all continuous maps from T to T'?

How to describe all continuous maps from $T'$ to $T$, where $T=\mathbb{R}$ with natural topology (base given by the intervals $(a,b)$ ), and $T'=\mathbb{R}$ with the topology with basis given ...
2
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0answers
270 views

Completely separated sets, Stone Cech compactification

This is a problem I'm confused with: Definition: We say two subsets $A$ and $B$ of a topological space $X$ are completely separated if there exists a continuous map $f: X \rightarrow \mathbb{R}$ ...
2
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0answers
182 views

On the Compact Uniformization Theorem

I just read through a proof of the Compact Uniformization Theorem, and I follow it up to the very last line. The proof is: Compact Uniformization Threorem. If $X$ is a compact regular space, then the ...
2
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0answers
155 views

How do physicists compute path integrals in Chern-Simons theory?

The space of connection has no measure right?
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45 views

Determining the interior of $([-1, 1]\times[-1, 1])\setminus \{ y \in \mathbb{R}^2 : d((0, 0), y) < 0.25 \} \subseteq \mathbb{R}^2$

Let $M = (\mathbb{R}^2, d_e)$ be the metric space, with $d_e$ the Euclidean metric. Let $C \subseteq \mathbb{R}^2$ be defined by $$C = ([-1, 1]\times[-1, 1]) \setminus \{ y \in \mathbb{R}^2 : d((0, ...
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0answers
36 views

Examples of generating the same topology

I'm teaching myself topology using a book I found. The question below is from the text. Then there are two additional questions that I am curious about. Please let me know if I'm doing it correct. ...
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21 views

A question uniform convergence

Let $X$ be a compact Hausdorff space, $a$ a continuous real-valued function on $X$, and for $t\in\mathbb{R}$ let $f_t(x)=\exp(ia(x))$ such that the function $t\mapsto f_t$ is continuous (where we use ...
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0answers
30 views

Sufficient condition for a infinite countable or non-countable intersection of open sets is equal to an open set.

Let $(X,\tau)$ a no discrete topological space. If necessary for an affirmative answer consider a metric space $(X, d )$ or a Banach space $(X, \|\,\cdot\, \|)$. In these cases, the topology $\tau $ ...
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0answers
37 views

Are these subsets open, closed, both or neither (revised)?

This is a follow up to Are these subsets open, closed, both or neither? Please let me know if my answers are correct, and If my reasoning is accurate and complete. Below are my corrections: ...
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0answers
39 views

Closed unit ball is a retract of $R^2$

I was asked whether a closed unit ball is a retract of the euclidean space $R^2$. I think the answer is yes and the retraction might be defined as follows: for all the points in $R^2$ join them with ...
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0answers
17 views

Covering a $n$-holed torus for $n\geq 2$ with a hyperbolic tesselation?

How can I cover a $n$-holed torus $(n\ge2)$ with $\frac{2-2n}{\frac pq-\frac p{2}+1}$ faces of regular hyperbolic tesselation {p,q}? I don't need the graphics, just the construction. For example, in ...
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0answers
31 views

How to cut a Fano plane and get a Mobius strip

Considering that the Fano plane is a finite projective plane, how can you cut it so as to get its constitutive mobius strip?
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46 views

Quotient spaces - $\Bbb R^1\hookrightarrow \Bbb R^3$

I am trying to understand quotient spaces, and I constructed my own example to do this: $(\Bbb R=\{(a,0,0)|a\in \Bbb R^1\}) \hookrightarrow (\Bbb R^3=\{(\alpha,\beta,\gamma)|\alpha,\beta,\gamma \in ...