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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454 views

Topological isomorphism vs isometric isomorphism

We say that: $T:(X,\|\cdot\|_X)\rightarrow (Y,\|\cdot\|_Y)$ is a isometric isomorphism if it is a linear isomorphism, and it is an isometry, that is $\|T(x)\|_Y=\|x\|_X\quad \forall x\in X;$ ...
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0answers
38 views

Canonical choice of inverse system for profinite set.

Let $X$ be a profinite set - an inverse limit $\varprojlim X_i$. How can one prove that then $X=\varprojlim Y_i$, where $Y_i$ is finite quotient spaces of $X$? I may prove it if $X$ is topological ...
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3answers
110 views

Is a direct proof of this possible

Consider the following statement $x_n \to x$ if and only if every subsequence of $x_n$ has a subsequence that converges to $x$. $\implies$ is clear. A proof of the other direction is given here. ...
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1answer
57 views

Simple question on connectedness in a subspace [duplicate]

For some reason I am having some trouble on this basic point set topology question: Suppose $X$ is connected, and $A$ is a connected subset of $X$, and that $B$ is a clopen set in $X-A$ (not in $X$, ...
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1answer
80 views

Munkres topology page 153

In Munkres p.153, we have a proof like this He mentions that $B_0$ is open, so there is some interval $(d, c]$ containing $c$, which is contained in $B_0$. So we know if $c=b$, $(d, c]=(d, ...
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1answer
64 views

To show closed subset of $R^2/{\sim}$ contains origin

I am working on the old preliminary exam from my university. I found trouble to solve the following problem. Could you please help me on it. Let $X = \mathbb R^2/{\sim}$ be quotient space where $x\sim ...
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1answer
167 views

Separability of the space of bounded uniformly continuous functions

Let $(X,\rho)$ a metric space. Do the space $U_b(X)$ of uniformly continuous and bounded real-valued functions on $X$ is separable? It seems that the point is to pass through the Stone-Cech ...
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0answers
33 views

Relation between $L^1(T)$ and $L^1[0,1]$

I know the question may be too general, but I need to know if there is a way in which I could relate the spaces $L^1(T)$ (where $T=\{e^{2 \pi i x}: x \in [0,1]\}$ and we use the Lebesgue measure on ...
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1answer
204 views

Gluing the ends of a cylinder. Can we get other than a torus?

Let $X=S^1 \times I$ be a cylinder, where $S^1$ is the 1-dimensional circle. If we glue the "bottom" boundary $S^1 \times 0$ and the "top" boundary $S^1\times 1$ by a homeomorphism sending $x\times 0$ ...
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1answer
84 views

Induced topology vs subspace topology

Reading my book I found this definition of an induced topology, which was then alleged to be equivalent to the standard definition of the subspace topology for that special case. However, I'm failing ...
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1answer
38 views

A question on metrizability

In Munkres, Topology, there is a theorem 10.3 Let $f:X\to Y$ Let X be a matrizable. The function f is continuous iff for every convergenct sequence $x_{n}\to x$ in X, the sequence $f(x_{n})\to f(x)$. ...
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1answer
104 views

Disconnected Topoological Space with Intermediate Value Property

Does There exist a disconnected topological space with intermediate value property? Intermediate Value Property states that 'a topological space X is said to have intermediate value property if for ...
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3answers
411 views

Let $X$ be a metric space with metric $d$. Show that $d:X \times X \longrightarrow \mathbb{R}$ is continuous.

Can someone please verify my proof or offer suggestions for improvement? I am aware that there is a similar question elsewhere, but I want help with my proof in particular. Let $X$ be a metric ...
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3answers
106 views

Nullhomotopic map extended

I have troubles understanding this proof: Let $h:S^1 \rightarrow X$ be a continuous map, then we have that if $h$ is nullhomotopic, $h$ can be extended to a continuous map $k:B^2 \rightarrow X.$ ...
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1answer
73 views

What is the definition of Lindelöf space?

My definition for "countable set" is a set with the cardinal $\aleph_0$ and "at most countable set" is a set $A$ such that $|A|≦\aleph_0$. Till now, my definition for Lindelöf space is a topological ...
3
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2answers
193 views

Proof of the Borsuk-Ulam Theorem

The Borsuk-Ulam Theorem says the following: For any continuous map $g: S^n \rightarrow \mathbb{R}^n$ there exists $x \in S^n$ such that $g(x)=g(-x)$. I'm trying to work through the proof given in ...
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0answers
43 views

Which part of differential geomety uses metrization theorems?

I learned three metrization theorems last year, which are Nagata-Smirnov,Smirnov and Bing. I thought these theorems are purely topological theorems, but i recently saw a post which says these ...
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4answers
3k views

Can a set be infinite and bounded?

I don't understand a statement in my math book course, I was restudying the compact sets part of the chapter when at a certain moment there is a corollary saying : 'every infinite and bounded part of ...
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1answer
32 views

Given Closed $U \subset C$, Closed $V\subset D$, $ C,\ D$ Both Closed,Show $U \cup V$ is closed in $C \cup D$

Given closed $U \subset C$, closed $V \subset D$, $C,\ D$ both closed. Show that $U \cup V$ is closed in $C \cup D$. Is the condition that $C,\ D$ being closed necessary here?
3
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1answer
68 views

Real analysis question about boundedness

In real analysis courses, students are often taught a theorem which states that: If $f$ is a real valued continuous function on $[0,1]$, then $f$ is bounded there and the example ...
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1answer
110 views

Axioms for a topology

I found this theorem in a book. I am not sure how I would prove it. To start with, I can't see why those conditions imply why the sets are open, nor even why they aren't closed (as for a finite set, ...
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1answer
118 views

Product of Borel $\sigma$-algebras vs Borel $\sigma$-algebra of product

If $X$ and $Y$ are topological spaces with associated Borel $\sigma$-algebras $\mathcal{B}_X$ and $\mathcal{B}_Y$, then the product $\sigma$-algebra $\mathcal{B}_X\otimes \mathcal{B}_Y\subset ...
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1answer
38 views

Fibre is open in covering space

I think I don't see the wood for the trees: In my notes I found the remark that if $p:E \rightarrow B$ is a covering map, then for each $b \in B$ we have that $p^{-1}(b)$ in $E$ has the discrete ...
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1answer
102 views

Homeomorphism of Interior of Convex Polygon to Open Unit Disk

Show that the interior of a non-degenerate convex polygon in $\mathbb{R}^2$ is homeomorphic to the open unit disk in $\mathbb{R}^2$. My attempt: Let $P$ be the set of points in the polygon, let ...
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1answer
46 views

If a topological space $S$ is second-countable, must necessarily every quotient space of $S$ be second-countable?

Let $S$ be a second countable topological space. Let $S^*$ be a quotient space of $S$ with quotient map $\pi$. If $\pi$ is open, it's easy to show that it transfers a basis of $S$ into a basis of ...
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1answer
649 views

Proof that X is connected if every continuous function on X has a fixed point

Just came from an exam and I am wondering how to prove the following: A topological space $X$ is connected if for each continuous function $f:X\rightarrow X$ there is a $x \in X$ such that ...
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1answer
65 views

Topological properties of $(0,1)\times \{0\}$

I am having a real hard time solving simple proofs involving open sets. I am confronted with this one: Is $(0,1)\times \{0\}$ open? Is it compact? What is its interior? I know $(0,1)$ is open. ...
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2answers
489 views

A topological function with only removable discontinuities

I've posted similar questions here and here, but no one has answered them to my satisfaction. Suppose that $f:\mathbb{R} \to \mathbb{R}$ is such that $\lim_{y\to x}f(y)$ exists for all $x$, that is, ...
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1answer
32 views

how we get a closure these set

I have some questions how we get a closure these set from any interval of real number For example (1,2),[3,6],(8,10],(2,infinity),(-infinity,4]? what is the formula closure in metric space?
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1answer
246 views

Is a Möbius Strip in > 4 dimensions impossible?

I seem to remember reading, on a plaque in the math building at Penn State, that Möbius Strips are only possible in 3 and 4 dimensions. In higher dimensional spaces, a Möbius strip will use the extra ...
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1answer
59 views

reason of defining continuous function between two topological spaces?

What is the reason of defining continuous function between two topological spaces ? (is it that under continuous function image of a compact/connected set is compact/connected)
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1answer
120 views

What does it mean for a group to act cocompactly by isometries on a topological space $X$?

What does it mean for a group to act cocompactly by isometries on a topological space $X$? I know if $X$ is a topological group, and $A$ a subspace, then $A$ is cocompact iff $X/A$ is compact. Not ...
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4answers
624 views

Prove that some topology is not metrizable

Suppose I am asked to show that some topology is not metrizable. What I have to prove exactly for that ?
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1answer
36 views

The configuration space of a compact space is not compact

Let $X$ be a hausdorff connected compact space and let $$C_3(X)=\{(x,y,z)\in X^3\;|\;x\not =y\not =z\}$$ And denote $\Delta$ the complement of $C_3(X)$ in $X^3$. I read that $C_3(X$ is not compact ...
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0answers
23 views

Prove that a continuous one-to-one function from a compact space onto a hausdorff space is a homeomorphism [duplicate]

Some lecture notes I'm reading use the following lemme: let $ f : X \to Y$ be a continuous one-to-one function from a compact topological space $X$ onto a hausdorff space $Y$. Then $f$ is a ...
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1answer
81 views

Homeomorphism between two subspaces

Let $X_1, X_2$ be subspaces given by $X_1 = (0,1) \cup (3,4)$ and $X_2 = (0,1)\cup(1,2)$. Are the subspaces homeomorphic? I basically tried proving the contrapositive, but that made it even more ...
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4answers
179 views

What is geometry, algebra, or topology? [closed]

I have trouble grasping the notion of geometry, algebra, and topology. An example is when someone might say, "I study the geometry of jet spaces" or "I study the lie algebras". What does it mean for a ...
2
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1answer
604 views

In a complete metric space with no isolated points, any countable intersection of open dense sets is uncountable?

I was playing with Baire's Theorem, and seemed to deduce the following: In a complete metric space $X$ that has no isolated points, any countable intersection of open dense sets is uncountable. ...
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1answer
41 views

Is Munkres referring to the topology generated by the basis here?

In the following lemma, is $\mathcal{T}$ necessarily the topology generated by the basis $\mathcal{B}$? I ask because it's not immediately clear to me, and also I don't see why the first sentence ...
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1answer
59 views

Is the homotopy class given by the degree?

Let $X$ be a topological space such that $\pi_n(X)=H_n(X)=Z$. A continuous map $f: S^n \rightarrow X$ is an element of $\pi_N(X)=Z$ therefore $[f]_{\mathrm{homotopy}}$ is characterised by an integer ...
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1answer
99 views

Fundamental group of the quotient of a cylinder by a rotation at either end

The following is a past qual problem: let $X = S^1 \times [0,1]$ be the cylinder, and define an equivalence relation on $X$ by $(z,1) \sim (iz,1)$ and $(w,0) \sim (e^{i\pi /7} w, 0)$. Compute ...
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0answers
262 views

Show that an hyperplane is closed iff f is linear and continuous

I need an help with the following exercise. Let $(E,\| \cdot \|)$ a n.v.s. and let $f:E\rightarrow \Bbb R$. Show that $H=\{x\in E: f(x)=\alpha\}$ is closed if and only if $f\in E'.$ Actually, I ...
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0answers
70 views

Suppose that $X$ is locally compact and $G$ acting on $X$ is proper. Show that the quotient $X/G$ is Hausdorff.

Suppose that $X$ is locally compact and $G$ acting on $X$ is proper. Show that the quotient $X/G$ is Hausdorff. I am working through some notes on Geometric Group Theory and I am having a hard time ...
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1answer
62 views

Maximal subspace on which an operator is bounded

Consider the Banach space $X=C[0,1]$ of real continuous function on $[0,1]$ equipped with the supremum norm. Consider the operator $A:D(A)\to X$, $Af=f'$ for each $f\in D(A)=C^1[0,1]$. We can see that ...
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2answers
715 views

Properties of reflexive Banach spaces

I just want to see the importance of reflexive Banach spaces and what is special about them compared to other Banach spaces. What kind of properties hold in reflexive spaces that do not necessarily ...
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1answer
96 views

Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces

Let's $(X_1,d_1), (X_2,d_2)$ be compact metric spaces such that for every finite subset of $X_1$ like $A$ (respectively any finite subset of $X_2$ like $B$ ) there exists a finite subset of $X_2$ ...
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1answer
93 views

Quotients of topological rings

Let $\varphi\colon R\to S$ be a surjective ring homomorphism and let $R$ be a topological ring. Is there some nice characterization of the finest topology on $S$ for with both $S$ becomes a ...
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3answers
108 views

Why does the coefficients of the expansion of (2x+1)^n produce the elements of the hypercube?

Why does the coefficients of the expansion of $(2x+1)^n$ produce the elements of the hypercube? For elements I mean the number of vertices, edges, square faces, cubes, hypercubes, etc that the next ...
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1answer
812 views

Reinventing The Wheel - Part 2: The Lebesgue Integral

Disclaimer After struggling for some time to find an appropriate definition for the notion of integration I came across another attempt for which I would need your help deciding to what extend this ...
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3answers
235 views

Prove that the unit disk is open

This isn't a homework question. For some reason, I'm having trouble proving this statement, even though it should be elementary. The question is: If we call the open unit disk $D$ (i.e., $D = \{ ...