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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65 views

How to organize my learning in Maths?

I m working on a problem in mechanics of material which concerns about the variation of shapes. I need to understand the deformation of material. I m a civil engineering graduate. All my understanding ...
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47 views

Determine the interior, boundary, exterior and closure of the set $S= \{(x_1,…,x_n)\in\mathbb R^n\mid \forall x_i\in \mathbb Q\}$

Determine the interior, boundary, exterior and closure of the set $$S= \{(x_1,...,x_n)\in\mathbb R^n\mid \forall x_i\in \mathbb Q\}$$ I´m using the following definitions: A set is closed if it is ...
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28 views

Let (X, T) be a topological space. If U is in T, do we use the notation for U is an element of T or can we also say U is a subset of T?

I know this is a basic question, and I am pretty certain that T, the collection of all open sets has only elements rather than subsets (at least not in this context). Could someone clarify? I know the ...
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18 views

Circle rotation number invariant under topological semi-conjugacy.

For a circle homeomorphism $f: S^1 \rightarrow S^1$ we can define the the rotation number $$ \rho(f) = \lim_{n \rightarrow \infty} \frac{1}{n}(F^n(x) - x) \mod 1, $$ for a lift $F:\mathbb{R} ...
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38 views

A homeomorphism preserves irreducible components?

Let $f$ a homeomorphism between two Hausdorff topological spaces $X$ and $Y$. Assume that $X$ and $Y$ are reduced analytic spaces. Is true that $f$ takes an irreducible component of $X$ in an ...
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30 views

differences between $Y^X$ and $C(X,Y)$ in topology

Can someone please explain me what are the definitions and differences between $Y^X$ and $C(X,Y)$ in topological spaces?
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28 views

About the conditions in Jordan's Curve Theorem

In the original formulation of the theorem, it was stated that a Jordan curve separate the plane in two sets that is not path-connected. The formulation in Wikipedia is that the Jordan curve separate ...
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49 views

How to prove a rectangle is compact

Show that the rectangle $K=[0,a]\times [0,b]$ is a compact subset of $\mathbb{R}^2$. My try : Take some open cover $\{U_{\alpha}\}_{\alpha\in \Lambda}$ of $K$. Now I tried to prove it is closed and ...
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30 views

Is the number of open sets same in homeomorphic topological spaces?

Does the homeomorphic topological spaces, have same number of open sets?
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34 views

The closure of $\{x\times0\mid 0<x<1\}$

I just want to know if I am right The closure of $\{x\times0\mid 0<x<1\}$ is $\{[0,1]\times0\}$ with the ordered square topology, is that right?
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30 views

Inner product spaces, normed spaces, metric spaces and topological spaces

I am collecting theorems or properties that hold in IPS, NS, MS or topological spaces, but not all of them. The reason is that I want to create some sort of overview over the respective spaces and ...
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42 views

The definition of the basis of the topology

The definition of the basis elements of the topology says that if x belongs to the intersection of two basis $B_1,B_2$,then $\exists$ a basis element $B_3$ s.t. $x\in B_3$. I am considering why we ...
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20 views

Show that $\operatorname{Int}(A)$ and $\operatorname{Bd}(A)$ are disjoint, and $\overline{A} = \operatorname{Int}(A) \cup \operatorname{Bd}(A)$

Can someone please verify my proof? If $A \subseteq X$, we define the boundary of $A$ by $\operatorname{Bd}(A) = \overline{A} \cap \overline{X-A}$. $\operatorname{Int}(A)$ is defined as the ...
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31 views

Def. about neighborhood on wiki

The def on wiki: "If $X$ is a topological space and $p$ is a point in $X$, a neighbourhood of $p$ is a subset $V \subseteq X$ that includes an open set $U$ containing $p$." And it says: "Note that ...
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31 views

Boundary of a holomorphic functions

Let $G ⊆ \mathbb{C}$ a bounded open connected set and let $f : \bar{G} → C$ a holomorphic function: Is this true? $$∂f(G) ⊆ f(∂G)$$ What I have is that $f$ is open.
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26 views

Error ? A subset $A$ of $ \mathbb R^p$ S.T $A^o = \phi$ and $A^- = \mathbb R^p$ where $A^o$ is interior of $A$ and $A^-$ is closure of $A$

Can there be a subset $A$ of $ \mathbb R^p$ such that $A^o = \phi$ and $A^- = \mathbb R^p$ where $A^o$ refers to the interior of $A$ and $A^-$ refers to closure of $A$ Attempt: By definition : $(i) ...
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42 views

Error? An open subset of $\mathbb R^p$ is connected if and only if it can be expressed as the union of two disjoint non-empty open sets.

I believe the book which I am reading has a printing error. One of the lemmas reads like this An open subset of $\mathbb R^p$ is connected $\iff$if it can be expressed as the union of two disjoint ...
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18 views

Natural surjection from complex upper half plane into modular curve

I am considering the natural surjection $\pi : \mathcal{H} \to Y(\Gamma)$ where $\mathcal{H}$ is the complex upper half plane and $Y(\Gamma)$ the modular curve of the congruence subgroup $\Gamma$. ...
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39 views

Disconnected Sets definition and connectedness of the unit interval

The definition of a disconnected set seems a bit ambiguous in the book I am reading : $1.$ A subset $D$ of $\mathbb R^p$ is said to be disconnected if there exist two open sets $A$ and $B$ such ...
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42 views

Darboux Integrable Functions vs. Uniform Closure of Simple Functions

Is there a Darboux integrable function $f:[0,1]\to E$ with values in a Banach space $E$ that is not the uniform limit of simple functions $s=\sum_\alpha\chi_{A_\alpha}$ taken over finite sums with ...
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23 views

Each infinite subspace of a KC-space …

A space in which all compact subsets are closed is called KC-space. A space in which every infinite set contains an infinite subset with only a finite number of accumulation points is said to have ...
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33 views

Find the point implied by intermediate value theorem

Consider a function $f(x)$ such that $f(0)=0$ and $$f'(x) = \frac{T-x}{T-f^{-1}(x)} + \frac{T-x}{S}$$ Then we can see that $f'(0)>1$ and $f'(T)=0$. Find $x$ such that $f'(x)=1$, in terms of the ...
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23 views

Intuitive affirmation on convex sets

Let $D_1, D_2$ two open, bounded and convex domain in $R^n$. Suppose that $D_2 \supset \overline{D_1}$, and the boundaries of these sets are of class $C^1$. Fix $x \in \partial D_1$ and suppose that ...
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16 views

Family of functions depending continuously on a parameter space WRT the $L^1$ norm

The material I'm reading involves a family of functions induced by a parameter space homeomorphic to an open disk. It attempts to show that the functions depend continuously on this parameter with ...
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43 views

A Local Homeomorphism Between Compact Connected Hausdorff Topological Spaces

Prove that a local homeomorphism between compact, connected, Hausdorff spaces is a covering map of finite degree. Attempt at solution: Let $f:M\rightarrow N$ be the local homeomorphism. Since $N$ is ...
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33 views

Homotopic maps of a compact polyhedron

My friend and I are trying to solve the following exercise. Problem: Let $X \subset \mathbb{R}^n$ be a compact polyhedron. Show that there exists $\alpha > 0$ such that for any pair of maps $f, g ...
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32 views

Can we call the boundary of a subset of a topological space “partial X”?

Intuitively, one might be tempted to say $\partial S$ (the boundary of $S\subseteq X$ for X a topological space) as "partial X". Is this formally valid?
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24 views

Paracompactness and partitions of unity

For Hausdorff spaces, paracompactness is equivalent to finding subordinate partitions of unity for any open cover. I am confused about the "easy" step. If $f_j$, $j \in J$ is a partition of unity ...
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25 views

Show that $\cup A_n$ is connected.

Can someone please verify my proof or offer suggestions for improvement? Let $\{A_n\}$ be a sequence of connected subspaces of $X$, such that $A_n \cap A_{n+1} \neq \varnothing$ for all $n$. Show ...
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31 views

Doubt about local flatness of low dimensional embeddings

I would like to know if it is possible to have a simple curve $\gamma $ on a surface $S$ such that $\gamma$ is compact and embedded (i.e. with respect to the topology induced from $S$ it is ...
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25 views

A local homeomorphism between compact, connected, topological spaces

Prove that every local homeomorphism between compact, connected, topological spaces is a covering map of some finite degree. If the spaces were Hausdorff, the proof is easy, since then the singleton ...
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17 views

How to call a function defined on a set with gaps on arbitrarily small scales.

Let $I$ be an interval and $A\subset I$ such that for any two points $x,x'\in A$ there exists an interval $J$ between $x$ and $x'$ such that $J\cap A=\emptyset$. How does one call this proerty of ...
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39 views

On Compact Open Topology

Consider $X$ as a compact topological space and $Y$ as a metric space. Consider $C(X,Y)$, the set of all continuous functions from $X$ to $Y$. Prove that $C(X,Y)$ with compact open topology is induced ...
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40 views

Proof: Product Topology Question XxY

If $f$ is maps from topogical spacce $Z$ to $X\times Y$ so: $f$ is continuous iff : $\begin{cases} (p_X)\circ f: Z \rightarrow X\times Y \rightarrow X \\ (p_Y)\circ f: Z \rightarrow X\times Y ...
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54 views

A Property of Baire Spaces

Let $X$ be a topological space. I define $X$ to have Property A provided that every closed meager subset of $X$ is nowhere dense. It is easy to see that all Baire spaces have Property A. Is the ...
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43 views

Completation of an n.v.s. and dimensions of subspaces.

I don't know if the following statement is true: Let $X$ be an n.v.s. with $\text{dim}(X)=\infty$ and not Banach; and $\bar X $ its completation in the bidual space. Let $Y$ be a closed subspace ...
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28 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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39 views

Proof about spectrum

Let X be a finite partially ordered set. How can to prove that there exists a ring R such that Spec R ≅ X? If anyone has any good way of thinking about them do please divulge..
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37 views

What does “single set” mean in this context?

I encountered this problem in Munkres topology. Let $X_1 , X_2$ denote a single set in topologies $\tau_1$ and $\tau_2$, respectively; let $Y_1 , Y_2$ denote a single set in the topologies $U_1, ...
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32 views

Urysohn's lemma and inf of rationals

In the course of the proof of Urysohn's lemma, one defines the function on the space X as the inf of a set of rational numbers that index sets containing each point x in X (except for f(F2) which is ...
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66 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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47 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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33 views

Hausdorff topology of a set of subsets

In the text I'm reading, there is a map from the C, the complex plane to E, a collection of compact subsets in C. Continuity with respect to the Hausdorff topology on E was talked about and I'm ...
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62 views

Confused by definition of an open set in “All the Mathematics You Missed”

On page 66 of Thomas Garrity's "All the Mathematics You Missed", Garrity gives the following definition of an open set in $\mathbb{R}^n$: A set $U$ in $\mathbb{R}^n$ will be open if given any $a ...
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30 views

Bounded Point in Uniform Spaces

I'm currently studying uniform spaces and have come across a problem I don't know how to solve. Given any vicinity $U$ of a non-discrete uniform space, I want to prove that for every pair of points ...
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13 views

Presheaf of real valued functions

Seen as how a Presheaf of real valued functions on a topological space X associates a function f:U→ℝ to each open set U, what function maps the empty set to ℝ since the empty set is by definition an ...
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47 views

Prove that a defined function g is continuous for a certain point

Consider the set of rational numbers $\mathbb{Q}$ equipped with the subspace topology inherited from $\mathbb{R}$. Let $c \in \mathbb{R}$. Define the function $g_{c}: \mathbb{Q} \to \mathbb{Q}$ via: ...
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46 views

Prove that $\mathbb{R} \times S^1$ is homeomorphic to $\mathbb{R^2} \setminus \{(0,0)\}$

I need to prove that $\mathbb{R} \times S^1$ is homeomorphic to $\mathbb{R^2} \setminus \{(0,0)\}$. I define the map $h:\mathbb{R} \times S^1 \to \mathbb{R^2} \setminus \{(0,0)\}$ by ...
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43 views

The set of rational numbers, each point is point accumulation

Please let us help someone by telling you a precise formulation is below, and then someone please tell me solution that has since become like that with a few days my friend we debates, here my ...
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37 views

Question about Boundary points of the sets in metric space

Let A be a metric spaces. Prove the following properties: The boundary of $A$ equals $A'-A$ The boundary of $A$ is the closed set. $A$ is closed if and only if it contains its boundary. Where ...