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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How to prove that the Topologist's Sine Curve is a non locally compact space?

Let $R^2$ have the Pythagorian topology. The subspace $Y=\{(0,0)\}\bigcup$ $\{(x,sin(1/x))$ |$ x>0\}$ is usually called the "Topologist's Sine Curve". I wish to prove that Y is a non locally ...
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17 views

Is there a model of set theory in which $2^{2^{\omega_1}}$ is separable?

We know that $2^{\mathfrak c^+}$ ($\mathfrak c =2^\omega=|\mathcal P (\omega)|$) is not separable by the following argument: Suppose $D$ is countable dense in $2^{\mathfrak c^+}$. For each ...
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38 views

Is the set of invertible upper triangular matrices open in $GL_n(\mathbb R)$? Is it open in the set of all upper triangular matrices?

I think the answer to the second question is yes, but can't quite prove this. I've no idea about the first part. I've done a few exercises of this kind but all have used the continuity of the ...
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2answers
33 views

For a compact set $K\subset \Bbb R^n $ prove the following :

For a compact set $K\subset \Bbb R^n $ and $\delta>0$ show that that there exists a finite number of elements in $K$, say $x_1,x_2,\dots,x_k$ such that any other element $x$ of $K$ is at a ...
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16 views

if $\Bbb B=\{x\in \Bbb R^{n+1}; \langle x,x\rangle<1\}$ be a open ball from Euclidean Space $\Bbb R^n$

I study Metric spaces and I has this problem Show that sphere $\Bbb S^n=\{x\in \Bbb R^{n+1}; \langle x,x\rangle=1\}$ is metrically homogeneous. For the other hands, if $\Bbb B=\{x\in \Bbb R^{n+1}; ...
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16 views

If C is a closed set in X and $x \notin C$, there is an open set $U$ containing C and $\epsilon > 0$ such that $U \cap B (x, \epsilon) = \emptyset$

I am trying to prove this statement. I have gotten to the point where I have $C$ to be a closed ball $\overline{B (y, \delta_1)}$ where I defined $U$ to be an open set $B(y, \delta_2)$ where $y \in ...
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22 views

Problem with topological space in probability theory. [on hold]

Let $(X, \tau)$ be a topological space. a) Show that arbitrary intersections of closed sets are closed. b) Prove that a set $F \subseteq X$ is closed if and only if for all sequences $\{x_{n}\} ...
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36 views

How to master general topology for analysis?

I started learning topology long ago. I first exposed myself to metric topology in Baby Rudin and Munkres Topology 2nd ed. Part I. Munkres is my most revisited book ever since. The first big ...
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2answers
27 views

prove that any continuous bijection from $S^2$ to itself is is an homeomorphism

Let $S^2=\{(x,y,z)\space:\space x^2+y^2+z^2=1 \}$ be a subspace of $(\mathbb R^3,d_{euclid})$. prove that every continuous bijection $F\space:\space S^2\rightarrow S^2$ is an homeomorphism from $S^2$ ...
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10 views

Proving convergence iff projection converges in product topology

This question is regarding the same problem. I wish to present my proposed solution and get feedback on my argument, and as such, I claim that it is not a duplicate. (In particular, the other asker ...
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1answer
24 views

What sequences could satisfy these requirements?

I need to find a sequence which converges to $0$ but is not in any space $\ell^p$, where $1 \leq p < +\infty$. And, I need to find a sequence which is in every space $\ell^p$ with $p > 1$ but ...
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37 views

Prove that $F$ is continuous iff for each $i$ $F_i$ is continuous.

let $(X,\tau)$ and $(Y_i,\sigma_i)\space i\in \mathbb N$ be topological spaces. let $Y:=\prod_{i=1}^\infty Y_i$ and $\sigma:=\prod_{i=1}^\infty \sigma_i$ the product topology (for any $U_1\times ...
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48 views

Let $ A\subset B$ be a closed and bounded set, and let $\sup(A)=b$. Show that $b \in A$.

Let $ A\subset R$ be a closed and bounded set, and let $\sup(A)=b$. Show that $b \in A$. I understand the concept but not quite sure where to begin for the proof.
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17 views

Searching for analytical or topological proof(s) of the Cayley-Hamilton theorem

Is there any analytical or topological proof(s) of the Cayley-Hamilton theorem ? I want to know such proofs ( if possible ) , I would even appreciate proper references with accessible links . Thanks ...
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23 views

Continuous maps vs. open maps [duplicate]

When we study topology, we typically study topological spaces and continuous maps between them. From a categorical perspective, this is "wrong," because continuous maps are not the structure ...
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21 views

Topology making a family of functions optimal

I am trying to do a problem in Arbib's Category Theory book. Loosely rephrased: Let $\{(X_i,\tau_i)\}_I$ be a family of topological spaces, $X$ a set, and $\{f_i:X\to X_i\}_I$ a family of functions. ...
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1answer
48 views

When is $x^{2^n}$ dense in $\mathbb{S}^1$, for $|x|=1$?

Motivation: So I just saw this question: Limit when $n\rightarrow\infty$ of $\text{sgn}(\sin(2^n \pi x))$ with $x\in(0,1)$ fixed., and the answer involves diadic numbers and things of the kind. Most ...
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1answer
30 views

Where was I supposed to use compactness in this proof that a compact subspace can be separated from a point by open sets?

Can anyone check my proof below? P. Let $X$ be a metric space. Prove that if $K\subseteq X$ is compact and $x\notin K$, there exist disjoint open sets $U$ and $V$ such that $K\subseteq U$ and ...
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1answer
50 views

How to conceptualize unintuitive topology?

I found Project Origami: Activities for Exploring Mathematics in my university's library the other day and quickly FUBAR'd (folded-up beyond all recognition) the couple sheets of paper I had with me ...
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12 views

Show that if $X_\alpha$ is Hausdorff for all $\alpha$, then $\prod X_\alpha$ is Hausdorff under the box and product topologies.

Can someone please verify my proof? I am aware that there is a similar question posted elsewhere, but I need help with my proof in particular. $\textbf{Note:}$ This is not homework. Show that if ...
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2answers
33 views

Are all the points in a nonempty open set limit points?

My conjecture is that given any open set $A\subseteq\mathbb{R}$, all points $a\in A$ are limit points. Prove, or if untrue, disprove by constructing a counterexample. A few definitions for ...
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18 views

Properties of normal set

For any closed subset $F$ of $X\subset \mathbb{R}^{n}$, we define the normal set $\mathcal{N}(F)$of $F$ as follows: if there exists $f\in C^2(X)$, and $x_0\in F$, such that $$ df(x_0)\neq 0;\\ ...
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32 views

Determine if the following set is open, closed or neither {$x\in\mathbb{R}^{2} | x_1 + x_2 = 1$} $\subset \mathbb{R}^{2}$

my textbook doesn't have solutions to most these problems, and this one is really giving me some trouble. Any help is appreciated. Determine if the following set is open, closed or neither ...
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24 views

$K$- Topology .aaa [on hold]

Define the $K$-topology of $\mathbb R$ as follows: Let $K=\{1/n | n\in \mathbb N\}$ and let $B_k$ be the collection of the subsets of the form $(a,b)-K$ in $\mathbb R$ and all open intervals. The ...
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1answer
32 views

Is the Riemann sphere homeomorphic to $\overline{\mathbb{R}}\times\overline{\mathbb{R}}$?

Let $\hat{\mathbb{C}}$ be the Riemann sphere. Let $\overline{\mathbb{R}}$ be the extended real. (i.e. $\mathbb{R}\cup\{\infty,-\infty\}$) Then, is $\hat{\mathbb{C}}\cong ...
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34 views

Why is this set connected?

I don't understand something about a proof of that if $X_i$ is a connected space for every $i\in I$, then $X=\Pi_{i\in I}X_i$ is connected. It is this: Let $x\in X$. Define $C$ the set of all $y\in ...
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1answer
26 views

Doubts about definition of open sets in “Understanding Analysis” by Stephen Abbott

In the book "Understanding Analysis" by Stephen Abbott, the author defines an open set as: A set $O \subseteq \mathbb{R}$ is open if for all points $a \in O$ there exists an ...
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28 views

How many Y shapes can you fit on the plane?

You can only fit at most countably many disjoint open discs on the plane: for any collection of disjoint open discs, it is possible to pick a single rational coordinate contained in each disc, and ...
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1answer
25 views

For $f:D\subset \Bbb R^n \rightarrow \Bbb R^m$ prove the following are equivalent:

For $f:D\subset \Bbb R^n \rightarrow \Bbb R^m$ prove the following are equivalent: a)$f$ is continuous in $D$ b)If $O\subset \Bbb R^m$$f$ is an open set, then there exists an open set $G\subset ...
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1answer
55 views

An open set in $\mathbb{R^n}$ is connected if and only if it is path connected

Here is a proof I found on the internet but cannot understand a part of it which is highlighted. I hope someone can help me understand this. Thanks in advance
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On distributivity of lattice of group topologies

Let $\frak L$ be the set of all topologies $\mathcal T$ on $\Bbb Q$ (the set of all rational numberes) such that $(\Bbb Q,\mathcal T)$ is a topological group. Then $(\frak L,\subseteq)$ is a complete ...
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1answer
22 views

Is there any special name for a $n$-torus made by products of hyperspheres?

I was wondering if there exist an accepted name for an $n$-torus made by the product of hyperspheres $\mathbb{S}^d$, that is for the following set: $$ ...
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30 views

Fundamental polygon of the sphere

Wikipedia shows me how the fundamental polygon of the sphere looks like, but I don't get this. I mean if you think of a sheet of paper and you glue A to A and B to B you get a triangle but not a ...
3
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1answer
48 views

When do functions turn a space into a locally ringed space?

Let $X$ be a topological space, and consider for each open set $U \subseteq X$ a set $F_U$ of functions $U \to k$ into some fixed field $k$. Let $\mathcal{O}$ be the sheaf of $k$-algebras induced by ...
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2answers
31 views

How to prove that $H(S_1\cap S_2)\subset H(S_1) \cap H(S_2)$ and $H(S_1 \cup S_2) \supset H(S_1) \cup H(S_2)$

I'm studying convex analysis and my task is to prove the following inclusions: $S_1, S_2$ are non-empty sets in $\mathbb{R}^n$, and $H(S) $ defined as the convex hull of set $S$. Show that ...
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31 views

Do two data sets have the same distributions?

$X = \{x_i\}_{i=1}^n \in \mathbb{R}^d$ is a data set, Whether there exists another data set $Y=\{y_i\}_{i=1}^n \in \mathbb{R}^m (d>m)$ so that the distribution of X and Y are approximately ...
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1answer
28 views

What does it mean “sequence with infinite range”

I'm trying to understand this phrase Find a sequence with infinite range that converges only to $0$. What does it mean "sequence with infinite range"? Thanks
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23 views

Show that if $F$ is continuous, then it is continuous in each variable separately.

Can someone please verify my proof? I am aware that there may be a similar question posted elsewhere, but I need help with my proof in particular. $\textbf{Note:}$ This is not homework. Let $F: X ...
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3answers
62 views

Proof of topological isomorphism

I remember reading in a section in plato.stanford.edu that the interval $(-∞, t)$ is topologically isomorphic to the interval $(0, t)$. I am not that good with topology, so could someone show me the ...
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Why are continuous functions the “right” morphisms between topological spaces?

Recently, someone mentioned to me that given a function $f: X \to Y$ there are two natural functions between the powersets $P(X)$ and $P(Y)$. Namely $f: U \subset X \mapsto f(U)$ and $f^{-1}: V ...
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1answer
21 views

Finding simple homotopy type

I have an excercise that I kind of dislike: Given $T-\{p,q\}$ where $T = S^1 \times S^1 $ and $p,q \in T$ two different points, I am supposed to find a simple homotopy equivalent space by ...
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37 views

Transitivity of smooth submanifolds

I was reading through Guillemin and Pollack and was having trouble verifying this for myself. Given $M \subset N$ and $N \subset P$, where $M$ is a submanifold of $N$, and $N$ a submanifold of $P$, ...
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17 views

Define the 4 types of interval subsets of the real numbers.

Define the 4 types of interval subsets of the real numbers. Is the union of an arbitrary number of open intervals also an open interval? Is the intersection of an arbitrary number of open intervals ...
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1answer
53 views

Example of sigma algebra that is not a topology

There is a very nice explanation of an example of sigma algebra that is not a topology: here. I do not fully understand the answer. Apparently this is a basic question, but why do we want this ...
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19 views

Is there a reason why M can't be all summable sequence?

Let M be the set of all summable non-negative sequences $\{x_k\}_{k=1}^\infty$ of real numbers, that is, $x_k \geq 0$ for all k and $\sum_{k=1}^\infty x_k$ converges to a real number. Let $d:M \to ...
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1answer
42 views

Can measure induce a topology on a Set?

When I was taught metric spaces in Topology, I came across the idea that metric defined on a set can induce a topology by creating a basis (open balls). If we have a measure defined on a set, can it ...
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2answers
51 views

What is $d(\sin(x),\cos(x))$ if d is a distance function in a metric space?

Let $M=\{f:[a,b] \to \textbf{R} | f \,is \,continuous \}$. Let $d:M \to \textbf{R}$ be defined by $d(f,g)=\int_a^b |f(x)-g(x)| \,dx$. What is d represent geometrically, and show that M, d is a metric ...
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58 views

Equivalence of Lebesgue Measurablity

Hello Mathematics Community. I am having some difficulties with the following problem dealing with Lebesgue Measure and its equivalent interpretation. I will first include the definitions which I am ...
2
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2answers
78 views

Why does the product topology allow proper subsets for only finitely many elements?

Consider Theorem 19.1 from Munkres' topology: The box topology on $\prod X_\alpha$ has as basis all sets of the form $\prod U_\alpha$, where $U_\alpha$ is open in $X_\alpha$ for each $\alpha$. The ...
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56 views

How to organize my learning in Maths?

I m working on a problem in mechanics of material which concerns about the variation of shapes. I need to understand the deformation of material. I m a civil engineering graduate. All my understanding ...