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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31 views

Linking of $S^p$ and $S^q$ in the $\mathbb{R}^d$ space

Can we have a nontrivial linking of a $S^p$ sphere and a $S^q$ sphere in the $\mathbb{R}^d$ space (or in the ${S}^d$ space)? I suppose that it can happen only if $p+q<d$. For example, we can have: ...
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12 views

Homology group and homotopy group of the standard twin

Given a 4-sphere, if we cut out a solid 3-torus $B^2 \times S^1 \times S^1$ from a 4-sphere $S^4$ (with an unknotted torus), the remained exterior is called "the standard twin," say $M$. What are ...
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52 views

Wot topology on $B(H)$ is not metrizable

Let $H$ be a infinite dimensional Hilbert space and $B(H)$ be the space of bounded and linear operators on $H$. I know that weak operator topology (wot) and strong operator topology (sot) are ...
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39 views

Marking Integers Using a Wheel

Suppose I had a wheel of diameter one meter and I was charged with marking every meter along an infinite stretch of a beach. The strategy is to insert pegs into the wheel so that every point that is a ...
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33 views

Order topology is regular and not normal

π-Base shows that linear order topology is not normal. But I remember in class the prof said order topology is normal. If $X$ is a set with linear order $<$, define a topology on X by letting ...
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31 views

$c_{00}$ is a dense subset of $c_0$

I would like to show that $c_{00}$ is a dense subset of $c_0$. I am not sure if I am overly simplifying the argument or even making the right argument for that matter. proof: Suppose that $x \in ...
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22 views

Characterizing equicontinuity via ultrafilters

We have a compact metric space $(X,d)$ and a homeomorphism $T:X\to X$. For any ultrafilter $p\in\beta\mathbb{Z}$ we can define the map $T^p:X\to X$ given by $T^p(x):=\lim_{n\to p}T^n x$ (which can ...
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40 views

$(X_1 \times X_2) \times X_3$ is homeomorphic to $X_1 \times (X_2 \times X_3)$

Let $X_1$, $X_2$, and $X_3$ be spaces. (a) Prove that $(X_1 \times X_2) \times X_3$ is homeomorphic to $(X_1 \times X_2) \times X_3$ is homeomorphic to $X_1 \times (X_2 \times X_3)$ So, I think I ...
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59 views

Finest good cover of a topological space

Let $X$ be a topological space. Does there exists a good open cover $\left\{ U_{a}\right\}_{a\in I}$ finer than any other open cover of $X$? A good cover $\{U_\alpha\}_\alpha$ of $X$ is a ...
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25 views

Two parallel lines with non-integer points identified: Is it a $T_1$ space?

I have a problem with an exercise. Let $Y$ the follow topological space of $\mathbb{R^2}$ with the euclidian topology. $$Y=\{(x,y) \in \mathbb{R^2}\mid y=0 \vee y=1 \} $$ And let $X=Y/\sim $ where ...
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60 views

Name of the set $B:= \overline{A}\setminus A$

Let $(X, \mathcal{T}_X)$ denote a topological space and let $A$ be a subset of $X$. We define the set $B:=\overline{A}\setminus A$. Does the set $B$ have a special name in the literature? All I could ...
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30 views

Hahn Banach theorem and supporting hyperplane theorem

The question is out of Rudin Functional analysis Chapter 3 problem 1. Call a set $H \subset \mathbb{R}$ a hyperplane if there exists real numbers $a_1,\ldots, a_n, c$ (with $a_i \neq 0$ for at least ...
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27 views

Identification of polygon edges

In Klein's famous example of regular 14-gon made of 336 copies of (2,3,7) triangles, he used identification for edges such that side 2i+1 is identified with side 2i+6 (mod 14). But I wonder how could ...
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28 views

Show that if $\forall$ $b\in B$, $f(b) \in T_y$ implies $f$ is open.

Let $f:(X,T_x) \rightarrow (Y,T_y)$ be a map between two topological spaces. Let $B$ be a basis for $T_x.$ Show that if $\forall$ $b\in B$, $f(b) \in T_y$ then $f$ is open. I am just looking for a ...
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26 views

Verifying a subbasis of the weak topology

So, I am working on a problem and I have shown that for $0 < p < 1$, $(\ell^p)^*= \ell^\infty$, I am now asked to show that the set of all $x$ with $\sum |x(n)| < 1$ is weakly bounded, but ...
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50 views

Showing the Sum of $n-1$ Tori is a Double Cover of the Sum of $n$ Copies of $\mathbb{RP}^2$

I want to show that the non-orientable surface of genus $n$ has a 2-sheeted cover by an orientable surface of genus $n-1$. The base cases are easy: $S^2$ covers $\mathbb{RP}^2$ and I worked on a ...
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41 views

Question on notation (topology & fiber bundles)

This is a very elementary question but I can't quite seem to track down a worthwhile source, so I was hoping someone more knowledgeable than I could lend their superiority. In Moore & Schochet's ...
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42 views

question on open connectedness in $\mathbb{R}^n$

My question is regarding an intermediary 'lemma' to deduce path-connectedness from an open connected set in Euclidean space. How does one prove that for any $x,y\in U \subset \mathbb{R}^n$, where $U$ ...
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54 views

An exercise on components of $\mathbb{S}^2$ as a closed combinatorial surface.

Suppose that the sphere $ \mathbb{S}^2 $ is given the structure of a closed combinatorial surface. Let $C$ be a subcomplex that is a simplicial circle. Suppose that $ \mathbb{S}^2\backslash C$ ...
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62 views

Aspect Ratio of Cylinder, Pyramid and Dome

The aspect ratio can easily be defined for rectangular geometries ($AR = height/width$). Is there a definition for aspect ratio of a dome, cylinder, and pyramid (Here standard pyramid and dome were ...
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35 views

Order and Metric

Consider the Reals as a totally ordered set via its natural order ( linear continuum ). Such order induces an order topology ( basis is the collection of open-intervals of that order), which ...
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59 views

If $E= A\cup B \cup C$ and $E$ is connected , where $A$ and $B$ are disconnected and $C$ is connected, then $A \cup C$ is connected.

If $E= A\cup B \cup C$ and $E$ is connected in a metric space $(X,d)$, where $A$ and $B$ are disconnected and $C$ is connected, then $A \cup C$ is connected. If we consider that $A \cup C$ is not ...
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67 views

Limit points in nonstandard analysis [solved]

Let $A\subseteq\mathbb{R}$, $p\in\mathbb{R}$. I proved that the following are equivalent: $\exists\left(x_{n}\right)_{n\in\mathbb{N}}\subseteq A\cap\left\{ p\right\} ^{c}$ such that ...
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111 views

Topology and Planetary Nebulae

I apologize ahead of time if this receives any down-votes, but I was just reading a text on topology when the idea struck me: has any mathematician or, for that matter, any topologist, analyzed the ...
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33 views

Connection between Chladni Plates and Algebraic Topology?

Does anybody know of a connection between Chladni Plates and Algebraic Topology? Any published research on the subject? I am interested to know if Chladni Plates (or surfaces) can be ...
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50 views

bounded continuous function are open in set of continuous functions?

Let $D$ be a metric space and $K \subset D$ a compact set. Let $C_b(K) = \{ f \in C(K, \mathbb{R}) | \mbox{ f is continuous and bounded } \}$. I want to prove that $C_b(K)$ and B(K,1) := $\{ f \in ...
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17 views

Basis and Subbasis verification

Suppose I have a set X = {a,b} and the topology T = {{a,b},{a},{b},{0}} where 0 is the empty set. Then a basis for the topology T is {{a,b},{b}, {a}} and also the subbasis is {{a},{b}}. Is any of this ...
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53 views

Producing $\mathbb{R}$ with countable amount of sets?

Prove, that you can't "produce" $\mathbb{R}$ with countable amount of sets, which are nowhere dense(I am not sure I said this definition correct, with nowhere dense, I mean that $Int(\overline X) = ...
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19 views

base of open neighborhood for dual group in k-topology

I wanted to ask the following: Suppose I have an abelian topological $G$, and $G^*$ is its dual group (all the continuous homomorphisms from $G$ to the circle group $T$). How can I show that the ...
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36 views

Topologizing ergodic system so that certain function becomes continuous

Let $X$ be a compact metric space and $\mathcal{B}$ its Borel $\sigma$-algebra. Suppose that $(X,\mathcal{B},\mu,T)$ is an invertible ergodic system ($T$ is only a measurable isomorphism, not ...
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50 views

Every Hilbert space is connected

Let $H$ be a Hilbert space. Proving $H$ is connected, suppose $\{e_i\}_{i\in I}$ is a orthogonal basis of $H$. Thus $H=\bigoplus_{i\in I} \Bbb C e_i$. Clearly $\Bbb Ce_i$ is connected for every $i$, ...
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31 views

Sequence of bounded sequence in metric space

I am reading a paper and bumped at this lemma which I do not know the proof and would like to see some reference. Please suggested me a possible reference. Let $M$ be a metric space and ...
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55 views

Compactness & Continuity - Looking for feedbacks on a specific setting

I am trying to get the implications of the following general setting concerning compact spaces and continuous maps. Any feedback would be greatly appreciated, because I have some difficulties in ...
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41 views

Is the function $f:(a,b) \to \mathbb R$ defined by $f(x):=\dfrac {x-(a+b)/2}{(x-a)(b-x)} , \forall x \in (a,b)$ a homeomorphism?

Is the function $f:(a,b) \to \mathbb R$ defined by $$f(x):=\dfrac {x-\dfrac{a+b}{2}}{(x-a)(b-x)} , \forall x \in (a,b)$$ a homeomorphism ? I have noticed that it is continuous and also noticed that ...
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40 views

connected and path connected

I have a discrete topological space consisting of the whole set the empty set and $\{0\},\{1\}$ .... I have shown it is not connected as both $\{0\}, \{1\}$ are clopen, however is this enough for me ...
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34 views

Counterexamples for the Converse of “Topological Conjugacy Implies Equal Topological Entropy”

Question: I would like to find two topological dynamical systems that are not topologically conjugate but nevertheless have the same topological entropy. Two topological dynamical systems $f:X\to ...
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77 views

Metrizable and First Countable Spaces

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. While studying Topological Spaces, I came across metrizable spaces. If I understand this ...
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24 views

Homotopy equivalence between O-O and $\theta$

Show that the dumbbell O-O (where there's no space between the "O" and "-") and the letter $\theta$ are homotopy equivalent, using the definition. So, let $X$ be the set of points in the dumbbell, ...
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64 views

Armstrong's “Basic Topology” proof of Tietze Extension Theorem is wrong - what's the best fix?

In Armstrong's "Basic Topology" he proves the Tietze Extension Theorem by first defining $d(x,A)$ (the distance from a point $x$ to a set $A$) as the infimum of a numbers $d(x,a)$ where $a\in A$. He ...
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42 views

Side identification on a hexagon

Apparently giving a hexagon side identification aabbcc results in a sphere. I'm struggling to see this, can someone explain? perhaps with a diagram? It seems to be all the vertices are identified, but ...
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102 views

Every order topology is regular (proof check)

My proof: Let $X$ be an space with the order topology, $x \in X$ and $F$ a closed set that does not contain $x$. Then, the set $X-F$ is an open set that contains $x$, hence there is an open set ...
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29 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 ...
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23 views

Minimal conditions for compactness of PDFs

I need to find some set of (minimal) conditions to put on a family of probability density functions with bounded support so that the family becomes compact. (I want to use Sion's theorem, which ...
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15 views

Dimension of a subgroup of a solenoid with measure zero

Let $G$ be a connected compact finite-dimensional abelian group (also called a solenoid). If $H$ is a subgroup of $G$ with Haar measure $0$, can we say something about the connectedness or the ...
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25 views

Non-trivial convergent sequences and continuous funtctions on toplogical spaces

Let $X$ and $Y$ be topoligical spaces. We say that $S\subset\ X$ is a non-trivial convergent sequence of $X$ if the following three conditions are fulfilled: i) ...
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30 views

How to make sure any two points with small enough distance are inside a common open set

Let $K$ be a compact subset of a metric space, and I cover $K$ with finite open sets. How can I select $\delta>0$, such that for any $x,y\in K$, with $d(x,y) <\delta$, $x,y$ are inside an open ...
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55 views

Homeomorphisms and Preorders on Topological Spaces

This is a follow up question to this question. It's not difficult,but I'm curious. Let X, Y be homeomorphic topological spaces and let ~ be a preorder on X. Then in general, wouldn't the relation be ...
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14 views

Question regarding uniform spaces and equicontinuity number 2

Following the already answered question: Question regarding uniform spaces and equicontinuity in the context of proposition 27. How do we know that indeed every element in the p-closure of G is a ...
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28 views

Dual group endowed with the compact-open topology

I wanted to ask a question. Let $G^*$ be the dual group of an abelian topological group $G$ ($G^*$ is defined to be the group of all continuous homomorphisms from $G$ to the circle group $T$). I ...
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24 views

Isomorphism of finite dimensional topological vector space with $(\mathbf{R}^k,\mathcal{R})$

Let $(T,\mathcal{T})$ be a topological vector space over $\mathbf{R}$ with finite positive dimension. Is it true that there exists an isomorphism between $(T,\mathcal{T})$ and ...