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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Prob. 5, Sec. 25 in Munkres' TOPOLOGY, 2nd ed: Is there a connected set that is locally connected at none of its points?

Let $A$ denote the rational points of the interval $[0,1] \times 0$ of $\mathbb{R}^2$. Let $T$ denote the union of all line segments joining the point $p = 0 \times 1$ to points of $A$. Then I can ...
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38 views

A problem about the intersection of convex open sets in $\Bbb R^m$

Let $X\subset \Bbb R^m$ be the union of convex open sets $X_1,\cdots,X_n$ such that $X_i\cap X_j\cap X_k\neq\varnothing$ for all $i, j, k$. Is $\bigcap\limits_{r=1}^nX_r\neq\varnothing$ true?
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22 views

Basis for a topology of a scheme

Suppose that $X$ is a proper and connected scheme over an algebraically closed field. Moreover let $\mathcal A$ be a collection of open subsets of $X$ with the following property: For every open ...
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29 views

Let $X=\mathbb Z^+$. Topologize $X$ with the finite complement topology. Let $A=\{3,5,6,7\}$. Find $\overline A$, int$(A)$, and bdry$(A)$.

Let $X=\mathbb Z^+$. Topologize $X$ with the finite complement topology. Let $A=\{3,5,6,7\}$. Find closure of $A$ $(\overline A)$, interior of $A$ (int$(A)$), and boundary of $A$ (bdry$(A)$). $A$ ...
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38 views

Shortest smooth paper Möbius Strip

I want to make a familiar Möbius strip of width 1 unit satisfying the physical properties of paper. Assume paper is a ruled surface, and the strip has to be smooth and non-self-intersecting. What ...
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31 views

A name for a particular covering map?

The quotient space of $\mathbb C$ obtained by identifying points differing by a Gaussian integer is topologically a torus. The map that takes each point in $\mathbb C$ to its corresponding point in ...
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72 views

Does the proof of productivity of connectedness require Axiom of Choice?

For arbitrary index set $\Lambda$, the product space $$ X = \prod_{\alpha \in \Lambda} X_{\alpha} $$ with product topology is connected if all of each $X_{\alpha}$ is connected. In the standard proof ...
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31 views

Is the boundary of a set a subset of the limit points?

Let $(X, \mathfrak T)$ be topological space and suppose that $A$ is a subset of $X$. Then $Bd(A) \subseteq A'$. My definition of boundary: Let $(X,\mathfrak T)$ be a topological space and let $A ...
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43 views

Path-components of the general linear group using only elementary algebra

Let $E(c)$ be an elementary matrix of the type to add $c$ times a row to another row when applied to another matrix on the left (with $c$ in some off-diagonal position $(i, j)$), and, with the usual ...
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17 views

Prove that the “additive” operation of the module($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) is continuous.

Consider the following module $\mathcal{M}=$($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) in which the "additive" operation is defined by normal multiplication in $\mathbb{Z}_{p}^{*}$ and scalar ...
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16 views

Finite intersection of arbitrary union not stable for arbitrary unions

It is a set-theoretic exercise to prove that the set of arbitrary unions of finite intersections of sets is still stable under finite intersections. However it is not true that finite intersection of ...
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32 views

weak closure of unitary group in $B(H)$

Let $H$ be a Hilbert space with dim $H=\infty$ , and $\cal{U}$ be the group of all unitaries on $H$. Show that the weak closure of $\cal{U}$ is a semigroup with respect to the multiplication. I know ...
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40 views

How to show a map is a homeomorphism?

I have calculated two of the properties of homeomorphism. Where I have found the bijective mapping and showed that $f$ is continuous. However i am not sure how to show that $f^{-1}$ is continuous?
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54 views

transformation of a folded piece of paper!!!

This is a question in the book Real Mathematical Analysis by Charles Chapman Pugh and I don't know how to face it! : Fold a piece of paper in half. (a) Is this a continuous transformation of one ...
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31 views

Quotient Space and Quotient Topology Definitions

I'm trying to show an equivalence between these two definitions: (1) The Quotient Space: Let $f: X \to Y$ be a map from a topological space $X$ to a set $Y$ and define $\pi: X \to \frac{X}{\sim}$ as ...
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25 views

A calculation for an open ball in $\mathbb{R}^N$ and function space.

Let $B_r(x)$ denoting the ball of center $x$ and radius $r>0$. We denote by $\lambda_{1,\,B_\rho(y)}$ the first eigenvalue of $-\Delta$ in $W^{1,\,2}_0\left(B_\rho(y)\right)$ and by ...
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36 views

quick question about a definition of homeomorphism set classes

Take $S$ a surface of general type. I want to define $Q$ the set of homeomorphism classes determined by the surface $S$. How can i define $Q$? I think that $Q$ is determined by all surfaces $S^{'}$ ...
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30 views

Lie bracket question

I am wondering if this is correct. Suppose $X$ and $Y$ are two smooth vector fields which vanish at $p$: $X(p) = Y(p) = 0$. Also assume that $[X, Y](p) = 0$. Is it true that the derivative of the ...
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47 views

Normal space is compact

I know that a compact Hausdorff space implies Normal, but does the converse holds? I.e. If a space is normal, it is compact and Haudorff. (Although $T_4$ imlicitly implies $T_2$)
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51 views

Hilbert Space is not locally compact.

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Show that Hilbert Space is not locally compact at any point. This is what I understand: ...
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47 views

Uncountability of $\mathbb{R}^I$ if $I$ is uncountable

Prove that if $I$ is uncountable, then $\mathbb{R}^I$ with the product topology is not countable. Based on what I have read, a set is uncountable if there is a bijection from that set to ...
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30 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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46 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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38 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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31 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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29 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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20 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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39 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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59 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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24 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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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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25 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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45 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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37 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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41 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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48 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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31 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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64 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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109 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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52 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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17 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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34 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 ...