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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5
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1answer
50 views

Compact topological space not having Countable Basis?

Does there exist a compact topological space not having countable basis? I have constructed a product space from uncountably many unit intervals $[0,1]$, endowed with the product topology. ...
0
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1answer
29 views

A simple question on Hausdorff distance

Let $(A_n)$ be a sequence of compact sets in $R^n$ and consider $K$ and $A$ compact sets in $R^n$. Suppose that $A_n \cup K \rightarrow A \cup K$ in the Hausdorff distance. Then $$ A_n ...
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0answers
35 views

Why is proof of the [topological] closed graph theorem incorrect?

Specifically, the closed graph theorem I am referring to is: Let $f : X \rightarrow Y$ exist and $Y$ be compact and Hausdorff. Then $f$ is continuous if and only if the graph of $f$ denoted by $G_f = ...
1
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1answer
54 views

Demonstrating that the Mandelbrot Set is connected

I know that demonstrating the Mandelbrot Set is connected requires a non-trivial proof, and that Mandelbrot himself was fooled at first. But can it be demonstrated visually that the set is connected? ...
3
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1answer
122 views

A proof about $F_\sigma$, $\sigma$-compact sets, and subsets of the irrationals

I've been looking at a proof that shows the following result. $\mathbb{P}$ is the set of irrational numbers, $\mathbb{Q}$ the rationals, and $\mathbb{R}$ the reals. The following conditions are ...
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1answer
24 views

Mapping Class Group of $S^3$

I am wondering if we can compute $\pi_0(Homeo(S^3))$ (i.e. the group of hoemomorphisms of the three-sphere mod isotopy) or if anyone has a reference where I could find such information.
5
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1answer
113 views

What exactly is a dimension?

Maybe this is too broad a question, maybe I need to be more specific. I am just clearing my head here, feel free to ignore at your pleasure. In Linear Algebra, we learned that the dimension of a ...
1
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1answer
30 views

Is the following proof of: $X = [0,\omega_1]$ does not satisfy $S_1(\Omega,\Gamma)$ correct?

Definitions: An open cover $\mathcal U$ of $X$ is called a $\gamma$-cover, if for every $x \in X$, the set $\{ U \in \mathcal U : x \notin U \}$ is finite. An open cover $\mathcal U$ of $X$ is ...
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1answer
37 views

being connected of R^infinity?

is $\mathbb R^\infty$ ($\mathbb R\times \mathbb R\times \mathbb R\times \mathbb R\times\cdots$) connected with metric $P$? $P(x,y)=\sup\{D(x(k),y(k))\mid k\in\mathbb N\}$ for all $i$: ...
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2answers
84 views

How is $\mathbb R^2\setminus \mathbb Q^2$ path connected?

Prove $(\mathbb R$ x $\mathbb R)-(\mathbb Q$ x$ \mathbb Q)$ is path connected. I know I need to let $(x_0, y_0), (x_1, y_1) \in$$(\mathbb R$ x $\mathbb R)-(\mathbb Q$ x$ \mathbb Q)$ and then consider ...
2
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1answer
23 views

Does parR imply Souslin?

I have encountered the following property in this article: We say that a space $X$ parR (partition-Rothberger), if, for every sequence $(\mathcal P_n : n \in \omega)$, of partition of $X$ into ...
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1answer
22 views

How to prove that any product of separable spaces has the Souslin property

A topological space $X$ has the Souslin property if every pairwise disjoint family of non-empty open subsets of $X$ is countable. I am trying to solve the following exercise: Prove that any product ...
2
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3answers
60 views

Is $\mathbb{Q}^2$ connected?

Is $(\mathbb Q \times \mathbb Q)$ connected? I am assuming it isn't because $\mathbb Q$ is disconnected. There is no interval that doesn't contain infinitely many rationals and irrationals. But ...
2
votes
1answer
23 views

Is set on lower-limit topology path-connected?

Is $\mathbb R$ endowed with the left-hand topology (also called lower limit topology) path-connected? Intuitively, I know that the answer is yes but I'm not sure how to prove it. Would it suffice to ...
2
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1answer
29 views

Is R with finite complement top path-connected?

I need to prove whether $\mathbb R$ with the finite complement topology is path-connected or not. Is the following proof valid? The function $ g:(\mathbb R,u) \rightarrow (\mathbb R,fc) $ is ...
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0answers
13 views

Action of Homeomorphisms on Proper Arc system.

Let $S_{g,n}$ be a surface of genus $g$ and with $n$ punctures. By an essential arc we mean an embeded arc (end points are in punctures) which is: Homotopically non-trivial i.e. not homotopic to a ...
3
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1answer
63 views

Size of topological space depending on the size of local basis. (With elementary submodels)

Recall that the character of a topological space $\chi(X)$ is the minimum cardinal $\kappa$ such that every point in $X$ has a local basis of size $\kappa$. I need to prove that if $X$ is compact ...
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3answers
60 views

Homotopy on the unit circle

I am trying understand why the identity function on the unit circle $X=\{(x,y): x^2+y^2=1\}$ is not homotopic to $f: X \to X$ where $f(z)=(1,0)$ for all $z\in X$.
0
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1answer
80 views

Prove that a closed ball is closed

Prove that in $\[\vec{E}\]$ normed vector space $\[B(\vec{x}, \varepsilon )\]$ is a closed set. and $\[B'(\vec{x}, r)\]$ is an open set. Fo the first part I created a sequence ($\[x_{n}\] $) ...
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4answers
46 views

Simply connectedness in $R^3$ with a spherical hole?

I understand why $R^3 - {(0,0,0)}$ is simply connected, and I also understand why $R^2 - {(0,0)}$ is not simply connected. The way I look it at is if checking if the region is $a)$ path-connected and ...
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0answers
34 views

Comparison of two final topologies

Consider the vector space $F$ of all infinite sequences of reals numbers, such that only finitely many terms of each sequence are nonzero. I recently encountered an exercise where I was required to ...
0
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1answer
35 views

Non First Countable: Lack of Information by Sequences [closed]

Can someone proof that when considering sequences in non first countable spaces information will be lost... I'm thinking of sth like there is necessarily a subset whos closure properly contains the ...
0
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1answer
29 views

problems with proving that f and g are homotopic.

i need to give an example of 2 continuous functions $f,g: X \rightarrow Y$ which are not homotopic, with: $X = [0,1] \times [0,1]$ and $Y = [0,1] \cup [2,3]$ and i need to show how many homotopical ...
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0answers
12 views

Calabi homomorphism of the disk

There is a fact that the homomorphism $Diff_0^{\infty}(\mathbb{D},\partial\mathbb{D},area)\to \mathbb{R}$ is surjective, we can use Calabi homomorphism to prove it, where ...
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2answers
31 views

strong topology = inductive limit topology on duals of projective limits

I've been bothering with this for some time now, and can't find any source with an actual proof, the statement simply appears to be "well-known". If you know (a source with) a proof, I'd be happy :) ...
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1answer
63 views

Need help understanding this proof about Gelfand spectrum

Consider the following theorem: Let $A$ be a complex non-unital commutative Banach algebra and let $\Omega (A)$ denote its Gelfand spectrum / character space. Then $\Omega (A)$ is locally compact. ...
0
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1answer
61 views

Is there a name for the one-point compactification of $\mathbb{C}$?

Let $\hat{\mathbb{C}}$ be the one-point compactification of $\mathbb{C}$. This space $\hat{\mathbb{C}}$ is called the Riemann sphere. If I want to designate the topology $\tau$ on ...
0
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1answer
36 views

What does 'real-valued' function mean in topology?

If I have a topological space $X$ and a 'real-valued' function $f$ on $X$. Does this mean I have a map of the form: $f: X \rightarrow \mathbb R$ where $\mathbb R$ has the usual topology? Or something ...
4
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2answers
400 views

How to think about a homeomorphism?

Two disjoint circles in the Euclidean space are homeomorphic to two circles interlocked without touching each other. My professor said that to a topologist they are the same thing. I don't understand ...
2
votes
1answer
40 views

Using Cantor's intersection theorem

Assume $f: X \rightarrow X$ is a continuous map where X is a compact metric space. Prove that there exists a non-empty set $A \subset X$ such that $f(A) = A$. (Hint: Set $F_1 = f(X), F_{n+1} = ...
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0answers
22 views

let $X$ be compact and $p:X\to X/R$. Prove that $X/R$ is hausdroff iff $p$ is closed.

Please refer me which book I should follow for answer. Let $X$ be compact and $p:X\to X/R$ prove that $X/R$ is hausdroff iff $p$ is closed.
2
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0answers
29 views

How to show that the Tychonoff product is associative?

Let $\{ X_t : t \in T\}$, be a family of topological spaces. Suppose thst $T = \bigcup \{ T_s : s \in S \}$, where $T_s \neq \emptyset $ for all $s \in S$, and $T_s \cap T_{s'} = \emptyset$ if $s ...
2
votes
1answer
42 views

Lie Automorphisms

Take $X$ to be a Lie group. Define a Lie automorphism of $X$ to be a group isomorphism from $X$ to itself which is also a homeomorphism. Define $Aut(X)$ to be the group of Lie automorphisms of $X$ ...
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0answers
20 views

What is the classification theorem of simple Lie groups?

I've seen this thrown around a bit, but I can't find what the theorem actually states? Can anyone help?
0
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1answer
29 views

Show that the interior of the set A is empty?

Consider $A = \{(x, \sin\frac{1}{x}) \mid 0< x \leq 1 \}$, a subset of $\mathbb R^2$. Find int($A$). We can see graphically that the interior of $A$ is definitely empty, but I want to check by the ...
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0answers
31 views

Seemingly simple question on Covering Property

I need to show that a closed, bounded set having exactly one accumulation point has the covering property. A set has the covering property if any open cover of it has a finite subcover. Since the ...
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0answers
27 views

Identification topology and disjoint unions

I was reading the book Basic Topology by M.A. Armstrong and I came across something I couldn't understand. I have uploaded the relevant pages- (1) What exactly is a disjoint union and what is the ...
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3answers
33 views

How is the function $g: \mathbb{R}_L \to \mathbb{R}$ defined by $g(x)=x\,$continuous?

$\mathbb{R}_L$ and $\mathbb{R}$ are the lower limit and standard topologies on $\mathbb{R}$. If $(a, b)$ is an open set in $\mathbb{R}$ then it's inverse, also $(a, b)$, isn't open in $\mathbb{R}_L$ ...
0
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1answer
42 views

Is $g(f(x,y),y)$ continuous for continuous $f,g$?

Suppose we have two topologically continuous (preimage of open set is open) maps $f:X \times Y \to Z$ and $g: Z \times X \to W$. Is the "composition", $h(x,y) = g(f(x,y),x)$ continuous? Why? My ...
3
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1answer
33 views

translation of open set problem

Suppose U is an open set in an Euclidean space. Then any point in U is contained in all but finitely many open sets that is translated by vectors converging to zero. It is easily proved in one ...
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0answers
45 views

Examples of topologies between norm and weak star

Let $X$ be a normed vector space and $X^\ast$ denote its continuous dual. The norm on $X^\ast$ is given by $\|\varphi\|=\sup_{\|x\|=1}|\varphi(x)|$. The weak star topology on $X^\ast$ is the weakest ...
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0answers
17 views

How to show that this two constructions of Tychonoff product topology are equivalent?

Definition: The Tychonoff product topology on $X = \Pi_{t \in T}X_t$, is the topology $\tau$, which is generated by the family $\bigcup\{ p_t^{-1}(\tau(X_t)) : t \in T \}$ as a subbase where ...
3
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1answer
14 views

Möbius bundle as a tautological bundle

I was trying to solve the exact same problem that was discussed in this question: Tautological vector bundle over $G_1(\mathbb{R^2})$ isomorphic to the Möbius bundle I came up with the step to ...
3
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2answers
22 views

Is an accumulation point of a set $A \subset C^*(X)$ always continuous?

Definitions: Let $C(X)$ be the space of continuous real valued functions of $X$, $C_p(X)$ be the space of real valued functions of $X$ with the topology of pointwise convergence. Denote $C^*(X) = \{ f ...
2
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1answer
32 views

Example where $f$ is discontinuous

Let $X,Y$ be topological spaces and $f: X \to Y$. I know that if $X,Y$ are not necessarily first countable (=countable nbhood base) then ''For all sequences $x_n\to x$ in $X$ it's true that $f(x_n) ...
4
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2answers
123 views

Show that $G$ ( subgroup of $\mathrm{GL}(E)$) is finite.

I came across with, I think, a difficult problem : Let E a Hermitian space with a Hermitian norm $||\ ||$. We provide $\mathcal{L}(E)$ with the norm $|||\ \ |||$ subordinated to $||\ ||$. ...
0
votes
1answer
37 views

Continuity of the sum of continuous functions

Let $X$ be a topological space and $f:X\to \mathbb{R}$ and $g:X\to \mathbb{R}$ be continuous functions. How do I show that $h:X\to \mathbb{R}$ where $h:=f+g$ is continuous, would prefer to use the ...
1
vote
1answer
24 views

Showing a simple Lie group is connected and compact.

I'm working on a presentation on simple Lie groups and would like to show by example that the simple Lie groups are connected, but I'm not really sure how to do this. I'd also like to show that one of ...
0
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1answer
27 views

Showing that a map can be deformed into the identity.

Suppose $F((a,b), k) = (ae^{\pi i k}, be^{\pi i k})$ where $0 \leq k \leq 1$. Now would $g(a,b)$ = $(-a,-b)$ if $g : S^{1} \rightarrow S^{1}$?
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1answer
27 views

Open quotient of second countable space is second countable

If $Y$ open and $X$ has countable basis, then $X/Y$ has a countable basis. This is a review problem for my midterm, and Wikipedia says it's true. However, I cannot find a simple proof anywhere. (At ...