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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58 views

What does $R^I$ stand for?

In section 30 of Munkres, one exercise states that "Give $R^I$ the uniform metric, where $I=[0,1]$". I guess it's not about powers or something, it's some conventional notation because I've never ...
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80 views

Lindelöf Property and Compact space

Let $X$ be a compact space and $L$ is the smallest family of subspaces of$\,X\,$that contains all closed sets and is closed with respect to countable union and intersection. The question is :- Is ...
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39 views

Levels sets of a continuous function

Suppose $f:[0,1]\rightarrow [0,1]$ is continuous. Let $A$ be the set of all maximal, connected subsets of the level set $f^{-1}(0)$. Can $A$ be uncountable?
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56 views

How is this topological space different from the euclidean one?

I'm preparing for my topology exam and came across this example which I can't figure out. Let $\mathcal{T}$ be a such family of all sets $U\subset \mathbb{R}^2$ that $U\cap L$ is an open set in L, ...
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23 views

Is the translation of open and closed sets to some language non-antonym preserving?

Maybe more than one person though, before you were given the definition of closed set, that they were the sets that are not open, i.e. that the property of open and closed being antonyms were ...
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106 views

Is this a general structure for constructs?

Here a construct is a category where the objects are sets and the morphisms are structure preserving functions. Common examples are groups, graphs and topological spaces. As far as I can see there is ...
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33 views

When is a metrizable topological vector space locally bounded?

Consider a topological vector space $E$ with topology $\sigma$. Suppose that $E$ is metrizable, in other words, that there exists a metric $d$ on $E$ that induces the topology $\sigma$. One can then ...
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38 views

An excerpt from a seminar

It is a statement that a professor made in a seminar which I attended yesterday.He says that the following hold: $1$.If $D$ denotes the closed unit disc then there does not exist a continuous ...
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16 views

path connectedness of space of almost commuting matrices

Let $R$ be a topological ring which is a domain. Let $n$ be an integer and let $\zeta_n$ be a $n$-th root of unity. Denote by $X$ the set of $m$ by $m$ invertible matrices with coefficients in $R$ ...
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89 views

Line with two origins is a manifold but not Hausdorff

The line with two origins is $(\mathbb{R} \times \{0,1\})/\sim$ where $(x,0)\sim(x,1)$ for $x\neq 0$. I can see that it is not Hausdorff, since we cannot separate the points $(0,0)$ and $(0,1)$. ...
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76 views

Problem 30 in the Exercises following Chapter 2 in Baby Rudin: How to immitate the proof of Theorem 2.43?

Here's problem 30, the very last one, in the Exercises after Chapter 2 in Walter Rudin's Principles of Mathematical Analysis, 3rd edition: Imitate the proof of Theorem 2.43 to obtain the following ...
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44 views

Condition ici=ic on a topological space is equivalent to if each dense set has dense interior in the space.

I am required to prove the following: Let $(X,\tau)$ be a topological space.Then each dense set has dense interior iff $ici=ic$ holds where $i$ is the interior operator and $c$ is the closure ...
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28 views

Proof of Triangulation Theorem for 1-Manifolds

While I am reading "Introduction to Topological Manifolds" by John M. Lee, I come to see the following paragraph in the proof of Theorem 5.10 pp. 102. Note that Int$\ e\cap\ $Int$\ e'$ is open in ...
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32 views

Is a countable, nowhere compact, zero-dimensional, dense in itself, Hausdorff space which is 2nd countable; homeomorphic to space of rationals?

Let $X$ be a countable, nowhere compact, zero-dimensional, dense in itself, Hausdorff space which is 2nd countable. Is $X$ homeomorphic to the space of rationals? $X$ is called nowhere compact when ...
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31 views

Is $S(\mathbb{R}^{d})$ dense in $L^{1}_{\textrm{loc}}(\mathbb{R}^{d})$?

Let $S(\mathbb{R}^{d})$ denote the class of Schwartz functions in $\mathbb{R}^{d}$. Is it true that $S(\mathbb{R}^{d})$ is dense in $L^{1}_{\textrm{loc}}(\mathbb{R}^{d})$, the locally integrable ...
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30 views

$A$ and $A+y$ are homeomorphic where $A$ is open set

Actually I need to understand $A+B$ is open whenever $A,B$ open set in $\mathbb{R}$ First I want to prove $A$ and translation of $A$ by $y,y\in B$ are homeomorphic $f:A\to A+y, f(x)=x+y$ may be the ...
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22 views

Singleton sets and net criteria for closeness

Theorem. Let $(X,U)$ be a topological space and let $A$ be a subset of $X$. Then $x \in cl(A)$ if and only if there is a net in $A$ that converges to $x$. My question? Does this theorem imply that ...
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38 views

Is this statement true?(covering map)

Let $C,X$ be topological spaces. Let $p:C\rightarrow X$ be a continuous function. Let $U$ be an evenly covered open subset of $X$. Let $V$ be an open subset of $C$ such that ...
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33 views

Showing $\phi(f \cdot g) = \phi(f) + \phi(g)$

For $\phi \in C^1(X; G)$ a cocycle being thought of as a function from paths in X to G, I want to show: $\phi(f \cdot g) = \phi(f) \cdot \phi(g)$. What I'm not sure is how I'm supposed to relate a ...
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39 views

Does there exist a continuous function between the following sets:

Does there exist a continuous function between the following sets: $A.f:(-1,1)\rightarrow (-1,1]$ which is onto and one-one $B.f:\{(x,y):y^2=4x\}\rightarrow \mathbb R$ which is one-one What ...
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62 views

Is the Zariski Topology

if $ K $ is an algebraically closed field, asks: Is there a point $ "w" $ of $ K ^ n $, is closed in the Zariski toplogy?
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46 views

Understanding the mechanics of P-adic topologies

I am trying to work out how it is that we actually work open sets on a p-adic topological space and how I would relate it to open sets in a point set topology. According wiki here: We have that open ...
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56 views

Lie group quotient structure

Let $G$ be a Lie group and $H$ a normal finite subgroup. Let $\pi : G \to G/H$ be the quotient surjection. How would one show that $G/H$ is a Lie group?
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34 views

Prove this map is continuous

$(rcos(t),rsin(t))↦((1/r).cos(t),(1/r).sin(t)), 0≤t≤2pi $ first for $0<r<1$, then for $r>1$ My idea is to say $(rcos(t),rsin(t)) = r .(cos(t),sin(t))$ then the cos and sin map with an ...
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23 views

Geometrical Explanation of Borsuk Theorem

Assume $K$, $L$ are $n$-pseudomanifold, and $K$ is compact. Let $f$ be a simplicial map between $K$ and $L$. We denote $n$-simplexes of $K$ and $L$ by $S_n(K)$, $S_n(L)$. Define ...
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22 views

Topology with equivalence of convergence of nets and almost everywhere convergence

I want to show that there is no topology for the set of Lebesgue measurable functions such that the net $<f_n> \to f$ iff $f_n \to f$ almost everywhere. Assume that there exists such a ...
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29 views

Helping me my study of introduction to analysis

I am a math major student who started study math now In my university class , my professor proposed me a few question and I thoought several hours but I can`t write logically so i ask about question ...
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32 views

When does a homogeneous space define a fibration?

Let $G$ be a locally compact and $\sigma$-compact group acting continuously and transitively on locally compact Hausdorff $X$. Then if $x_0 \in X$ and $H_{x_0}$ denotes the isotropy group at $x_0$ we ...
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45 views

Uniqueness of the universal covering space (up to an isomorphism)

Let $Y_1$, $Y_2$ be universal covering spaces of some topological space $X$. I want to show that $Y_1$ are $Y_2$ are isomorphic. Denote $p_1 \colon Y_1 \to X$, $p_2 \colon Y_2 \to X$ the projections. ...
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36 views

Existence of covering space

I would like to know that if $X$ is a connected topological space, there is always a covering space of it, i.e., a continuous map $p:X'\to X$ with the known property.
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35 views

Union over disjoint union

How does the normal union behave over the disjoint union? For instance, if i have some indexed collection of disjoint unions between two sets, what is the union over the whole collection?
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43 views

Continuous Strong-Strong Implies Continuous Weak-Weak

Let $X$ and $Y$ be two Banach spaces and let $T$ be a linear map between $X$ and $Y$. Show that $T$ is continuous strong-strong if and only if $T$ is continuous weak-weak. I can see that $T$ ...
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46 views

Proving an attractor (i.e set with self similarity) is connected

Let $K$ be an attractor for iterating function system of two similarity maps i.e $$K=f_1(K)\cup f_2(K)$$ A similarity map is defined to be $f_i:\mathbb{R}^d\to \mathbb{R}^d$ s.t $$\forall x,y\in ...
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33 views

Dual of a locally convex space

Let $X$ be a normed space. Suppose $E$ is a subset of $ X^*$ (The space of continuous linear functionals). For every $\phi\in E$, define seminorm $p_\phi: X\to [0,\infty)$ such that $p_\infty (x)= ...
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20 views

very elementary question about bases on the real line

Let $\mathcal{U} = \{ (-\infty,a) : a \in \mathbb{R} \} $. I want to show $\mathcal{U}$ is a basis for a topology on the real line. Attempt Let $x \in \mathbb{R}$. Choose $I = (-\infty, x+1)$. Then ...
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72 views

Topological entropy of isometric extension

L.s., This is a homework question some of my fellow students and I are having great difficulty with. Let $Y,Z$ be compact metric spaces, $X = Y \times Z$, and $\pi$ the projection to $Y$. Denote $h$ ...
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54 views

Dictionary order topology and subspace topology

Compare $(0,1) \times (0,1)$ with the dictionary order topology to the same set with the subspace topology given by the dictionary order on $\mathbb{R} \times \mathbb{R}$. This is an exercise in my ...
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52 views

highway metric topologically equivalent to euclidean metric?

Consider the Euclidean metric space $(S, d_1)$ on $\mathbb{R^2}$ and the highway metric space $(S, d_h)$ on $\mathbb{R^2}$, where the highway metric is defined as $$d_h(x,y) = \begin{cases} |x_2 ...
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100 views

Connected sum of orientable manifolds

I was reading through Lee's Smooth Manifolds on the part regarding orientations and I was wondering if the connected sum preserves the orientability of manifolds. Intuitively it seems to be true, but ...
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33 views

Question considering the covering map of the circle $S^1$ in Munkres, 2. edition

The map $p: R \rightarrow S^1$ given by the equation $p(x) = (\cos 2\pi x,\sin 2\pi x)$ And we are to consider the subset U of $S^1$ consisting of those points having positive first coordinate. Then ...
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98 views

Confirm solution to chapter 2, Problem 18 in Rudin's book: principals of mathematical analysis

Is there a non-empty perfect set $E$ in $\mathbb{R}^1$ which contains no rational numbers? My effort: Yes, there is. We take $E_0 \colon = [\sqrt{2},\sqrt{3}]$. Then $E_0$ is non-empty, closed, ...
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67 views

Set of limit points for the set of all integers.

If the set of all limit points are in the set $E$, then $E$ is a closed set. Suppose that I want to show that the set of all integers is a closed set. Is it right to say that since the set of all ...
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21 views

Doubt in a step of the proof of Rado-Kneser-Choquet theorem

I am trying to prove Rado-Kneser-Choquet theorem, which states that if $f$ is sense preserving self homeomorphism of the unit circle $\partial D$. Then harmonic extension $F$ of $f$ is self ...
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68 views

Sets, Topology and Applying Cantor's Intersection Theorem

I am trying to solve the problem related to the Sierspinski triangle. The triangle is shown as follow. Let $S$ be the intersection of all the finite stages a). Show that $S$ is a nonempty compact ...
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29 views

Cut sets in topology

As part of a problem sheet I have been asked to show that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$ whenever $n \neq m$. When I first proved that $\mathbb{R}$ is not homeomorphic to ...
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66 views

Homotopic attaching maps give Homotopy Equivalent spaces

I want to prove that if $f,g : S^{n-1} \to X$ are homotopic maps then the resulting spaces $X \cup_f D^n$ and $X \cup_g D^n$ are homotopy equivalent. I know this question has been asked before: ...
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19 views

Continuity definitions on non-compact subsets

At the top of the wikipedia articles on Hölder condition, Lipschitz Continuity and others, the chain of logic is as follows: On a compact subspace of a metric space: $$Continuously Differentiable ...
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55 views

Interior of sum of sets equals sum of interior of summands

I'd like to have the answer to the following question. If $X_1,X_2\subseteq \mathbb{R}^n$ are convex and compact sets of dimension $n$, does the following hold: ...
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45 views

Proper Maps: Where is continuity used in this Wikipedia proof?

In this article on Wikipedia, a proof is given of the statement that any map $f$ from $X\to Y$ that is closed, continuous, and has the property that $f^{-1}(\{y\})$ is compact in $X$ for $y\in Y$, is ...
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33 views

condition for homeomorphism

If $X $ and $Y $ are homeomorphic as topological spaces is there any necessary and sufficient condition for $X\setminus A$ and $Y \setminus B$ to be homeomorphic?$ A\subseteq X ,B\subseteq Y$