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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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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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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74 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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73 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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43 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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25 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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31 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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21 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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61 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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45 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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32 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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21 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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29 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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42 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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32 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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40 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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71 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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51 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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42 views

Continuous map between topological spaces

I'm trying to prove that the following map between two topological spaces is continuous. Any help? $$f: ([0,1] \times [0,1], T) \to (S^1 \times S^1, T')$$ $$(x,y) \mapsto ((e^{2i\pi x}, e^{2i\pi ...
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48 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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83 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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91 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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64 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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18 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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62 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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27 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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59 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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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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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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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$
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51 views

$A$ is an interval so $A$ is connected?

I want to prove that if $A\subset \mathbb{R}$ is an interval then $A$ is connected. I found this proof, and I don't understand it essentially the ii) Suppose that $A$ is an interval but not ...
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25 views

How do you show Euler characteristic of any convex polyhedron is $2$?

In the Euler characteristic proof of a convex polyhedron, how do you show the cell decomposition of projection of two polyhedra 1) have a common refinement AND 2) that common refinement comes from ...
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136 views

Borel Measures: Atoms (Summary)

Disclaimer: The question here has been solved, now: Finest Measurable Partition (For jeapardy it is stated below, anyway. Have fun! ;) ) Summary: This is a summary of the discussions: ...
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47 views

Tangent Bundles to manifolds

I am having trouble trying to visualize exactly what a tangent bundle to the klein bottle is spuposed to look like. Is it possible for one to decompose it as a direct sum of simpler bundles?
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53 views

Textbooks to complete concurrently - Self learning empowerment

A user is completing some year challenge that takes them through $9$ textbooks and they are alternating in author. Algebra - Cohn Analysis - Rudin Topology - Lee Repeat three times. I would like ...
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37 views

Separated Spaces and a Partition Differences?

I am just getting a handle on separated definitions from Topology , reading Munkres. So the definition of a separated subsets of a topology, is that they are both disjoint. Further, if each subset ...