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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34 views

Equivalence on pseudocompactness.

Does someone know a reference or proof of the following equivalence on pseudocompactness? A completely regular space $X$ is pseudocompact if, and only if, every non empty $G_\delta$ set in $\beta ...
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1answer
38 views

Class $C^1$ with “open” domain?

In my book, a function is of class $C^1$ if it's defined on an open domain and ... (and then the rest follows in regards to the derivatives), but what if it's defined on a closed domain? So then it ...
2
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1answer
130 views

How to prove if a set is bounded

I'm looking through my real analysis text book, and I KNOW, that this particular set is bounded: The set of complex numbers Z, such that |z| is less than or equal 1. I know that it is contained in a ...
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1answer
33 views

A problem about covering map and evenly covered neighborhood

Suppose that p : E -> B be a covering map and that B is path-connected and locally path-connected. Then, given y in B, there exists a neighborhood U of y in B that is path-connected and evenly covered ...
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1answer
79 views

Is the following set compact

Let $S$ be the set $S = \{(e^{-x}\cos (x), e^{-x}\sin(x)) : x\geq 0\} \cup \{(x,0):0\leq x \leq 1\}$. Is $S$ compact? I know I have to show that if $S$ is closed and bounded, then it is compact. ...
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84 views

Why is this map homeomorphism? [duplicate]

Munkres - Topology p.143 Let $p:X\rightarrow Y$ be a quotient map. Set $x\sim y$ iff $p(x)=p(y)$. Let $X/\sim$ be the quotient space of $X$ with respect to $\sim$. Let $\pi:X\rightarrow X/\sim$ be ...
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1answer
108 views

Neighborhood base of weak topology

If $X$ is a Banach space, the weak topology on $X$ is the weakest topology in which each functional $f$ in $X^\ast$ is continuous. I have some difficulties in understanding its neighborhood basis in ...
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53 views

Example of a proper metric space such that the associated length space is not proper.

Can anyone give me a example of a proper metric space $(X,d)$ such that $(X,\bar d)$, where $\bar d$ is the induced length metric, is not proper. I have a example but I am not sure if it is right. ...
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1answer
40 views

Is there an equivalence relation of which the quotient topology is homeomorphic to codomain of a quotient map?

Let $X,Y$ be topological spaces. Let $p:X\rightarrow Y$ be a quotient map. Then, does there exist an equivalence relation $\sim$ on $X$ such that $X/\sim$ is homeomorphic to $Y$?
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1answer
86 views

A few questions on $\omega_1$

I'm trying to understand several things regarding $\omega_1$ and trying to get a better feeling of it. First question is - is $\omega_1$ a connected space? I think it isn't, but not really sure how ...
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1answer
93 views

General topology proof exercise

Let $f:X\to Y$ be a function between topological spaces. Prove that, if: $$(A \cap cl(B))\cup(cl(A)\cap B)\neq \emptyset \implies (f(A) \cap cl(f(B)))\cup(cl(f(A))\cap f(B))\neq \emptyset $$ ...
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1answer
44 views

Proof of closed graph theorem with nets

For my topology class, I have to prove that if $X$ and $Y$ are compact Hausdorff and the graph of $f:X \to Y$ is closed, then $f$ is continuous. If $\{x_\lambda\}$is a net in $X$ that converges to ...
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2answers
38 views

Homotopy inverse and retracts

I was thinking about the following. Let $f:X \rightarrow Y$ and $g:Y \rightarrow X$ be a homotopy equivalence. I was wondering about the induced maps on the fundamentalgroups and whether we have ...
22
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1answer
346 views

Does a set $A \subseteq [0,1]$ exist such that $A$ is homeomorphic to $[0,1] \setminus A$?

Does a set $A \subseteq [0,1]$ exist such that $A$ is homeomorphic to $[0,1] \setminus A$? I have no idea how to attack this problem. Any help will be appreciated.
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3answers
144 views

Undergrad level presentation on homological algebra and some related topics

I'm a TA of an introductory course about modules, category theory and homological algebra and the students have to do a 2 hour long presentation as a final exam. There's one student who really likes ...
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1answer
54 views

Proper map not closed

We call a map proper if the inverse image of quasi-compact sets are quasi-compact sets.(Add the Hausforff axiom if convient) Is there example for such a map not be a closed map? (If the spaces are ...
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0answers
53 views

Face post of a subcomplex complement

Let $P$ denote the face poset of a simplicial complex, $\Delta$ the order complex of a poset, and $\simeq$ homotopy equivalence. It's known that for any finite simplicial complex $\mathcal{K}$ that ...
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1answer
80 views

Some problems from section 4 of Munkres

I'm right now covering Section 4 of Topology by James R. Munkres, 2nd edition, and am stuck with the following problems in the exercise set after Section 4: Problem 8(c): Show that given $a$ with ...
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2answers
123 views

Topology: Boundary of boundary

How to prove that for any $S\subset\mathrm{R^\mathrm{n}}$, $\partial\partial\mathrm{S}\subset\partial\mathrm{S}$ , using the concept of 'open ball'? Progress For every open ball on any $x \in ...
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0answers
100 views

Definition and Intuition of a Weakly Dense Set

What does it mean to say: set A is "weakly dense" in a set B? The definition of a "dense set" is rather intuitive: the classic example of Q (rationals) being dense in R (reals) is very clear. How ...
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1answer
41 views

Two conditions for continuity?

Below, $f$ is a function $X\rightarrow X'$ between topological spaces. Are both those conditions equivalent with continuity? $x\in (f^{-1}(A'))^i\Rightarrow f(x)\in (A^{\prime})^i$, for any ...
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3answers
70 views

Real analysis questions ( 2)

This probably will look like I'm trying to get you to answer my homework, but I'm not. All I'm looking for is to understand the problem and concepts involved with the problem. Here is the problems. ...
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0answers
42 views

Norms on $\mathbb{R}$ seen as a $\mathbb{Q}$-vector space.

this is not really a question : I had some ideas on topics I don't feel secure with. I expose these hereafter : are there any mistakes in my reasonning ? Also, if anyone knows a good read about this ...
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1answer
58 views

Is cone not a topological manifold?

Is the cone = X a Hausdorff, second-countable topological space that is not a topological manifold? Since the open subsets $U_{\alpha}$ do not cover the vertex of the cone, so $U_{\alpha}$ is not a ...
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102 views

Proving that the set of real numbers is a topological space.

I recently finished an activity provided by a professor where one of the questions was to prove that the set of real numbers is a topological space. The hint provided was to "consider the union of ...
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1answer
120 views

Do there exist space-filling curves that fill the whole plane? If so, can they be visualized?

I know that there are space-filling curves from $[0, 1]$ to the unit square, and this question addresses curves transforming the real line into the entire plane. But what about transforming the unit ...
2
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1answer
71 views

A question about the Skorokhod topology

I have a question which may be naive but I can not find the answer in general reference about Skorokhod topology. Let $\{w_n\}_{n\ge 0}$ be a sequence of cadlag functions defined on $[0,1]$ such ...
3
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1answer
52 views

The existence of a simply-connected neighborhood of a contractible loop

Let $M$ be a smooth manifold with a point $x_0$ on $M$ and a smooth loop $\gamma$ at $x_0$. If $\gamma=0$ in $\pi_1(M,x_0)$, then can we find a simply-connected open set $U$ around $x_0$ such that ...
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0answers
128 views

Nets and convergence of Riemann sum

STATEMENT: Let $[a,b]$ be a finite interval in $\mathbb{R}$. Let $\Lambda$ be the collection of all finite subsets of $[a,b]$ that contain $a$ and $b$. When orderedf by inclusion, it is a directed ...
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1answer
68 views

$K$-theory for non-compact manifolds

It seems that usually (by which I mean, in every source I've looked at) people define the group $K^0(X)$ for $X$ compact Hausdorff. Sometimes they later extend this definition to all locally compact ...
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2answers
71 views

Euclidean topology on $\mathbb{R}^{m+n}$ is equivalent to the product topology on $\mathbb{R}^m \times \mathbb{R}^n$

I'm attempting to teach myself topology for graduate school this summer, but I'm having a tough time. I'm trying to prove that the Euclidean topology on $\mathbb{R}^{m+n}$ is equivalent to the ...
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1answer
39 views

Continuity of inverse for the uniform convergence

Let $X$ be a compact metric space, and denote by $G$ the group of all homeomorphisms of $X$ endowed with the topology of uniform convergence. Is it true that the inverse mapping $h \mapsto h^{-1}$ is ...
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1answer
38 views

An Exercise from Manetti's Topologia

I Have to show that that the quotient space $R^2/D^2$ is homeomorphic to $R^2$. My idea is to find a continuous function $$ f: R^2 \mapsto R^2 $$ such that $f(D^2)=O$, where $O=(0,0)$. Nevertheless, I ...
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0answers
48 views

$f:D\subset \Bbb R^2 \rightarrow \Bbb R$, where $D$ is a compact and convex set, reaches it maximum at $int(D)$

I'm trying to prove that if $D$ is a compact and convex (for every two elements of $D$, the line that connects them is contained in $D$) then: If $f:D\subset \Bbb R^2 \rightarrow \Bbb R$ and at ...
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1answer
46 views

Prove that there always exists a ray contained in $K$

Let $K$ be a convex, closed and unbounded set in $\mathbb{R}^n$. Show that for any $x\in K$ there exists a ray ($\{x+tv:t\ge0\}$ with some $v\in\mathbb{R}^n\setminus 0$) contained in $K$ with start ...
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1answer
45 views

Is $\mathcal{B}_a$ non empty?

I'm having trouble understanding exactly what the basis for a system of neighborhoods is, so I thought asking this question might help me understand. if $a\in X$ and X is a metric space, is ...
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3answers
46 views

Obtaining the usual metric on $\mathbb R$ by “pulling back” $\arctan$

This is a part from an e-mail that a professor wrote to me. I couldn't understand what he meant by Consider two different metrics on the real line $\mathbb R$ inducing its usual topology: one is ...
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0answers
45 views

Conceptual proofs to seven color theorem of torus for 17-19 year olds

what is the best way to explain the seven color theorem of torus to some high school kids and freshman college people?
3
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1answer
41 views

Non Hausdorff complex manifold?

Suppose there is a topological space $X$ equipped with an atlas making it into a complex manifold. Is it true that this implies automatically that $X$ is Hausdorff ? I have heard it is the case for ...
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2answers
74 views

Topology as a generalization of the open sets of the reals

As far as I can make out, topology is a generalization of the properties of open sets of the reals; this is evident in the terminology; for a set $X$, subsets of this space are actually called open if ...
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39 views

question about Skorokhod distance

Let $D:=D([0,1], R)$ be the space of all cadlag functions defined on $[0,1]$. Now we have the known Skorokhod topology defined by: $\forall f, g\in D$ ...
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1answer
36 views

Uniform limit of real valued functions on a compact space. Is the union of their images necessarily compact?

Let $K$ be a compact space with $f_n$, $f$ continuous functions $K \to \mathbb{R}$ such that $f_n \to f$ uniformly. Is $\mathrm{im}f \cup \bigcup \mathrm{im}(f_n) \subseteq \mathbb{R}$ necessarily ...
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1answer
29 views

Proof Verification : If $S$ is a metric space and $U(x)$ is the component of $S$ containing $x$, then is $U(x)$ closed in $S$ .

If $S$ is a metric space and $U(x)$ is the component of $S$ containing $x$, then is $U(x)$ open in $S$? Attempt: $U(x)$ is the union of all connected subsets of $S$ containing $\{x\}$. Hence, $U(x)$ ...
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1answer
158 views

Prove $\mathscr B$ is a base of a topology

Let $X = \mathbb Z$ (the integers), and for each $n \in \mathbb Z$, define $B_n = \{m \in \mathbb Z: m \geq n\}$. Let $\mathscr B = \{B_n : n \in Z\}$. Prove $\mathscr B$ is a base for a topology on ...
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1answer
31 views

Is $S^1$ Hausdorff with respect to induced topology from $\mathbb{R}^2$

I am not sure if this is correct: Is $S^1 = \{ (x,y):x^2 + y^2 = 1\}$ Hausdorff with respect to induced topology ($f:X\to Y$, the induced topology on $X$ by $f$ is $\{ f^{-1}(U): U \in ...
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2answers
145 views

Is the set of rationals between $\sqrt{2}$ and $\sqrt{3}$ open or closed in $\mathbb Q$?

Consider the set of all rationals, $\mathbb Q$ as a subset of the set of all reals $\mathbb R$. Assign $\mathbb Q$ the subspace topology induced by the standard topology on $\mathbb R$. ...
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3answers
60 views

topology induced on Y by f?

I don't think I understand the definition of induced topology very well, since I got confused trying to prove it's a topology. I have this definition: $f:Y \to X$, $\mathcal{T^x}$ topology on $X$ ...
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3answers
86 views

Disproving a statement about connectedness.

If $X \subset M$ is connected and $M$ is a metric space with $d(y,z) = 2$, for $y,z \in X$. Must there be an $x \in X$, such that $d(x,z) = 1$? I know I am supposed to get a contradiction, but I ...
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1answer
86 views

Continuous functions on a Suslin line

This question is motivated by Brian Scott's answer in this thread. It looks to me that continuous functions on Suslin lines may have remarkable properties (from my perspective). Convention. I am ...
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2answers
41 views

Comparing definitions of Continuity given in Real Analysis and Topology respectively

In Real Analysis we define continuity at a point, using the epsilon-delta formulation, however we encounter a more general definition of continuity in topology. Can we use modify that definition for ...