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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Relation between the Effros structure and Vietoris topology

Let $(X,\tau)$ be a topological space (eventually, a polish space) and $\mathcal{F}$ the collection of all closed sets of $X$. Given $\mathcal{U}\subseteq \tau$ finite, define $$ ...
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28 views

Projective limit involving p-adic numbers

Let $p$ and $q$ be distinct primes. What is the projective limit $$\varprojlim \mathbb R^2 / (p^n \mathbb Z \times q^n \mathbb Z)?$$ That's an exercise from Robert's book on $p$-adic analysis. Is it ...
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16 views

Thms on Dynamical Systems: Cont. functions on Compact Spaces (sources)

I've recently started taking a discrete-time dynamical systems perspective on a topic. I've been able to introduce a reasonable metric on a set, obtaining a compact space. Under that metric, a nice ...
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55 views

Why is this countable union of closed sets closed?

I'm having trouble understanding the logic of the following statement (taken from pg 53 of Lebesgue Integration on Euclidean Space by Frank Jones): "Though unions of countably many closed sets are ...
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36 views

open cover definition of compactness - technicality I don't understand

Let $(X,d)$ be a metric space. Let $A$ be a subset of $X$. $A$ is compact if out of every open cover $\{D_{j}\}_{j \in \mathcal{J}}$ of $A$, there exists a finite subcover (that is to say, there ...
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72 views

Local boundedness of monotone operators in general spaces

A classical result states that: If $X$ is a Banach space then every multi-valued monotone operator $T:X\to 2^{X^*}$ is locally bounded on $\operatorname*{int}D(T)$ (the interior of its domain). I ...
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16 views

What is unilateral continuity?

I wanted to ask if anyone knows what unilateral continuity is. In particular, I am considering the multiplication map of group from $G\times G$ to $G$. This is being used in one of the papers I am ...
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55 views

The number of Connected Components of a topological space

Let $X$ be a topological space, and $Y$ a closed subset of $X$. If we can express $Y$ as a finite disjoint union of connected closed subsets of $X$, is this expression unique or at least the number of ...
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29 views

Definition of locally connected metric space

I have this definition of locally connected metric space: "A metric space $(X,d)$ is called locally connected if for all $x\in X$ and for all $U\subset X$, $U$ neighbourhood of $x$, exists a connected ...
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24 views

Quintic equation and number of lines on the quintic

I heard a talk where the speaker said that the solution to the equation $x_1^5 +x_2^5 +x_3^5 +x_4^5 +x_5^5 = 0$ is a six-dimensional (Calabi-Yau) manifold. Then he went on to define five curves of ...
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26 views

Distinctions of different topologies on the sequence space (countable cartesian products of $\mathbb{R}$)

$\newcommand{\b}[1]{\mathbf{#1}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\N}{\mathbb{N}}$ Question I solved this exercise in Munkres.(20.4) But I don't know if I did it righ t or not. I really ...
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45 views

Show that there exist Borel sets $B_n$ such that $B=\bigcup B_n$

Let $X$ be a Polish space. Let $B$ be a Borel subset of $X \times X$ with the following property: $$\forall x \in X \ \left|\left\{y : (x,y) \in B\right\}\right| = \aleph_0.$$ Show that there exists ...
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40 views

Are the sections of entourages in a uniform space open?

Wikipedia's article on uniform spaces defines the following. A nonempty family $\mathcal{U}$ of subsets $U \subseteq X \times X$ is a uniform structure if it satisfies the following axioms: ...
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50 views

Closure and interior of subspaces

a) Let $X = C_b(\mathbb{R})$ thse set of bounded continuous functions on $\mathbb{R}$. We have the subset $Y = \{f \in X : f(t)= f(t+1)$ for all $t \in \mathbb{R} \}$. $Y$ is closed relative to the ...
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23 views

Checking proof that space is not path connected

I was going through Munkres and ran into a problem asking to show that \begin{align*} X & = \operatorname{cl} \left( \left\{ \left( x, \sin \frac{1}{x} \right) : 0 < x \leq 1 \right\} \right) ...
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16 views

Induced topology on a set by the image of open sets

If $X$ is a topological space, $Y$ is a set, and $f:X \rightarrow Y$ is a map, when is possible to create a topology for $Y$ with the images of open sets in $X$? This is not always possible, because ...
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49 views

Prove the product topology is the equivalence of metric topology in a special case.

Define a topology T on D$^\Bbb N$ with D={0,1} and the discrete metric as the following: C $\in$ T is basic iff there's an i $\in \Bbb N$ such that $C := C_1 \times C_2 \times...\times C_j \times D ...
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43 views

An exercise related to Krull topology - showing that two bases define the same topology

The following is a definition from my lecture notes: Let $L/K$ be a Galois extension with $G=Gal(L/K)$ then the family subgroups $Gal(L/L_{i})$, where $L_{i}$ runs over all finite ...
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50 views

Segment ordered density conjecture revisited

I have a set $S\subset\mathbb {R}^2$ with the following property (P) $\forall x,y\in S$, $\forall\mathscr{C}$ a convex set that contains $x$ in its interior, $bd\mathscr{C}\cap [x,y]\subset ...
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16 views

How do we look at a metric and determine what the neighborhoods look like?

We were given some examples of metrics in class: $l$-metric: $d(x,y)= \sum_{i=1}^n |x_i-y_i|$ (open diamonds) $\infty$-metric: $d(x,y)=\max |x_i-y_i|$ (open squares) $p$-metric: $d_p(x,y)= ...
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29 views

Prove that there does not exist a homotopy for a space (similar to Topologist's Comb)

Suppose $X$ is the subspace of $\mathbb{R}^2$ consisting of straight-line segments joining $(0,1)$ to the points $(1/n,0)$, for $n\in\mathbb{N}$ and the segment joining $(0,1)$ and $(0,0)$. This space ...
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11 views

A surface is essential iff each component of it is essential?

First, for the terms, A surface is essential if it is both incompressible and boundary-incompressible. I want to show that A surface S is essential if ans only if each components of S is essential. ...
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26 views

Sphere as a countable collection of Circles

The obvious answer is to use the Baire Category Theorem and show that it is impossible to write $S_2$ as the countable union of Circles. I wish to present a more intuitive, straight forward, argument ...
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44 views

Finiteness of Lusternik-Shnirelman category

Are there conditions on a topological space $X$ under which its Lusternik-Shnirelman category is countable (or even finite)? "Countable Lusternik-Shnirelman category" means that $X$ can be covered by ...
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25 views

Links in $\mathbb R^3$ up to continuous isotopies.

It is well-known that any knot in $\mathbb R^3$ is isotopic to the unknot, if we are in the continuous category, i.e. if isotopy is only through continuous embeddings. This is so because the "knotted" ...
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27 views

Digesting a proof of Bieberbach's theorem on crystallographic groups

Here is a proof of Bieberbach's theorem: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.460.1436&rep=rep1&type=pdf I can follow the computing details, but I have no intuitive sense ...
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37 views

Necessary and sufficient condition for a topological space to be Hausdorff.

Problem: Show that $X$ is Hausdorff if and only if the diagonal $\Delta=\{x\times x|x\in X\}$ is closed in $X\times X$. Attempt: Suppose $X$ is Hausdorff. Then to show $\Delta$ is closed, we need to ...
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57 views

If two total spaces are locally homeomorphic to their base spaces, when will a fibre preserving map between them be continuous?

Let $fp = p'g$, where $f$ is continuous, $p$ and $p'$ are local homeomorphisms. When will $g$ be continuous ? My reading says it is iff any local (continuous) section $s$ of $p$ over $U$ which is ...
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Show that $||x||$ is a norm on $\mathbb{R^n}$

Let $A\subset \mathbb{R^n}$ be any limited, open, convex, and the centre symmetry set having centre at 0. Show that $||x|| = \inf \{k>0 : x/k \in A \}$ is a norm at $\mathbb{R^n}$ and open ball ...
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145 views

Prove the tangent space at a point $x$ of the $n$-sphere is the space $\{v \in \mathbb{R}^{n+1} : v\cdot x=0\}$

I can see why this is true but I'm not sure how to prove it, any help would be appreciated. Prove that the tangent space $TS^{n}_{x}$ at a point $x$ on the $n$-sphere $S^{n}:=\{x \in ...
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37 views

Is preimage of a point in a compact Hausdorff space under a continuous closed surjection compact?

If $X$ is compact Hausdorff and $p:X\rightarrow Y$ is continuous, closed, and surjective, can I say that for any $y\in Y$ that $p^{-1}(y)$ is compact in $X$. Ultimately I'm trying to prove that $Y$ ...
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Small objects in TOP

Are spheres and disks small objects in TOP (more precisely, small with respect all continuos maps)?
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$A\cup B$ and $A\cap B$ locally connected.

Let $X$ be locally connected, $X=A\cup B$, where $A,B$ are closed and $A\cap B$ is locally connected. Prove $A$ and $B$ are locally connected. Let $x\in A$. Then, either $x\in A\setminus B$ or ...
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55 views

Compactification of sine curve

The title may be a little inaccurate because part of my issue is that I can't make complete sense of what the following question is asking, let alone how to go about solving it: Note that if ...
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39 views

Definition of One-point compactification

My question is an elementary one but I can't seem to find the formal definition. If $M$ and $N$ are topological spaces and $\dot{M}$ is the one-point compactification of $M$ then how is a map $f:M ...
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19 views

Closed Subset of $l^{1}$ Space

Let $l^{1}$ be the space of real sequences with the usual metric. Consider the sequence $(x_{n})_{i=1}^{\infty}$ where we define for $x_{i}$, $x_{i}^{n}=1+\frac{1}{i}$ if $n=i$ and $x_{i}^{n}=0$ ...
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68 views

When is the countable intersection of open sets a closed set?

When is the countable intersection of open sets a closed set? (I.e., what are sufficient conditions for the closedness of the intersection?)
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Example of continuous bijection between homeomorphic spaces that is not a homeomorphism?

I know this question has been asked before, and marked as a duplicate of: Are continuous self-bijections of connected spaces homeomorphisms?, but I didn't understand the example of the zipping pants ...
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15 views

Existence of initial topology wrt family of maps?

Let $((X_i, \mathcal{O}_i))_{i\in I}$ be a family of topological spaces, $X$ a set and $(f_i:X\longrightarrow X_i)_{i\in I}$ a family of maps. The initial topology determined by the family ...
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Does the set of isolated singularities not have limit point?

Let $G$ be open in $\mathbb{C}$. Let $f:G\rightarrow \mathbb{C}$ be a function. Define $D=\{z\in G: f \text{ is complex-differentiable at } z\}$ Assume that $z$ is an isolated singularity of $f$ ...
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The rotation map between $S^n$

First question, which maps we have from $\mathbb{S}^n$ to $\mathbb{S}^m$(if $m$ is not equal to $n$ )? The rotation or another mapping? Ok, how to express this mapping. Second, let $z$ be any element ...
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Is this map well defined?

Let $X$ be a connected, locally path connected, locally compact metric space and $G$ be a group of homeomorphisms of $X$. Let $\bar{G}$ be the closure of $G$ in Homeo$(X)$ endowed with the compact ...
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49 views

Maximal torus of a Lie group.

I would like to ask the following question: Let $G$ be a (simple if needed) connected Lie group and $T$ 'the' maximal torus of G. suppose now that G is endowed with an other Hausdorff topology $\tau$ ...
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Does every nontrivial quotient $\mathcal{H}/\Gamma$ have an unramified cover

If $\Gamma$ is a proper finite index subgroup of $PSL_2(\mathbb{Z}) \cong C_2*C_3$, then must there exist a $\Gamma'$ finite index in $\Gamma$ such that ...
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42 views

How do I prove this net has such point?

Let $X$ be a compact connected Hausdorff space and $M$ be the set of compact connected subspaces of $X$. Let $\mathscr{C}$ be a nonempty chain in $(M,\subset)$. Let $a:\mathscr{C}\rightarrow \bigcup ...
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28 views

To characterize large separable spaces in which every non-trivial connected subset has non-empty interior?

$\mathbb R$ with usual metric is a space such that every connected subset with more than one point has non-empty interior . My question is , can we characterize those separable metric spaces with more ...
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Why is the immersed image of the Klein bottle $S^2$ with three closed discs identified?

Someone said that there are three discs glued together on the Klein bottle self-intersecting itself in a circle. Here is a picture: They said that one from the "near" side of the tiny connecting ...
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50 views

Prove that there is a metric on the product space $\mathbb R^{\mathbb N}$ relative to which $\mathbb R^{\mathbb N}$ is complete.

Prove that there is a metric on the product space $\mathbb R^{\mathbb N}$ relative to which $\mathbb R^{\mathbb N}$ is complete. A hint has been given to take the metric D(x,y)= $=\sup ...
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72 views

Definition of $\Delta$-complex: restriction to interior is homeomorphism

From Hatcher's Algebraic Topology, the definition of a $\Delta$-complex is: Then, a page or so later, he writes: Regarding the sentence "Condition (iii) then implies...", I don't understand why ...
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60 views

Show that $S^n$ is homeomorphic to $(D^n\times \{-1, 1\})/{\sim}$, where $(x, 1)\sim (x, -1)$ for all $x$ in boundary of $D^n$

I understand some basic examples of homeomorphisms such as Show that $\mathbb{R}^2/{\sim}$ is homeomorphic to the sphere $S^2$. but I don't know where to start with this example. Do I have to ...