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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1answer
16 views

Comb space is contractible but not base point preserving

For each positive integer $n$, let $I_n=\{1/n\}\times I$ as a subset of $I\times I$. Let $X=(I\times0)\cup(0\times I)\cup(\bigcup_{n\geq 1} I_n).$ Let $x_0=(0,1)\in X$ be the base point. Show that $X$ ...
-2
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1answer
50 views

How to prove that a set is an open set [on hold]

$X=]0,1[$ How do I prove, that $X\subseteq \mathbb R$ is an open set? How do I prove, that $X\subseteq \mathbb C$ is not an open set? This is supposed to be possible to prove by using only the ...
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0answers
23 views

title for proposal in general topology [on hold]

i have some problems by writing my proposal in master hence i want to ask an opinion about the title for the proposal,i'm planning to do about the general topology. But i'm still confused to create my ...
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2answers
29 views

For which compact sets can the size of the finite subcover be bounded?

I've been struggling to find a solution to this problem: For which compact sets can you set an upper bound on the number of sets in a subcover of an open cover. My understanding is that I need to ...
0
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0answers
16 views

If every infinite subset of $S$ has an accumulation point in $S$, then $S$ is bounded

If every infinite subset of $S$ has an accumulation point in $S$, then $S$ is bounded. Proof: Suppose $S$ is unbounded. then , for every $m >0,~~\exists~~x_m \in S$ s.t. $|x_m|>m.$ The ...
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0answers
19 views

Explain the Concept of “endedness”

Particularly spaces that are one-ended, two-ended, ... $k$-ended. Can anyone explain via simple examples? Also why two spaces with different ended-ness are not isomorphic.
7
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0answers
47 views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
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0answers
14 views

How to fold a 3-sphere

I have seen in a few texts about simple topology some methods for constructing 2D surfaces by folding a square patch. The edges of the patch are given arrows, much like the n-headed parallel symbols ...
2
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3answers
26 views

Analysis proof with metric spaces

Part a) of Theorem 2.27 in baby Rudin reads (roughly) as follows: Theorem. Let $X$ be a metric space and $E\subset X$. Then the closure of $E$ is closed. Proof. Let $x\in\bar{E}^c$. Then ...
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1answer
23 views

If a collection is locally finite, then the collection of all closures is also locally finite

I want to prove the following Lemma. Let $A$ be a locally finite collection of subset of a topological space $X$. Then the collection $B=\{\bar{a}\mid a\in A\}$ of the closure of elements of $A$ is ...
2
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1answer
36 views

Topology of a manifold

A manifold $M$ is a locally euclidian topological space (every point has a neighborhood homeomorphic to an open subset of $\mathbb{R}^n$). We assume, in addition, $M$ Haussdorf and second countable. ...
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1answer
13 views

If $S$ is a subset of $\mathbb R^p$, then every infinite subset of $S$ has an accumulation point in $S \implies S$ is closed

If $S$ is a subset of $\mathbb R^p$, then every infinite subset of $S$ has an accumulation point in $S \implies S$ is closed. My query is : Isn't the above statement self proving? Every infinite ...
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1answer
19 views

multiplication of compact sets

There are $ A $ and $ B $ subsets of $ \mathbb{R} $, defined $ AB = \{ ab: a \in A, b \in B\} $. Now suppose that $ A $ and $ B $ are compact sets, then prove that $ AB $ is a compact set. I took a ...
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0answers
35 views

Question on topology and Zorn's lemma

I am having trouble showing a paracompact cover has a local refinement (that part is by definition) which admits another cover indexed by the same set such that each open set in the new set has ...
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3answers
76 views

How does one think in terms of the definition of compactness?

(noob question; I'm rather confused by the compactness definition and haven't found any help on the internet yet)... So the definition given in my textbook for compactness is: a set X is compact if, ...
0
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1answer
53 views

From metric to topological vector space

Suppose that $E = C[0,1]$ and suppose we have a metric given by $$d(f,g) = \int_0^1 \min(|f(x)-g(x)|,1)dx$$ Why is it that the topology defined by this metric makes $E$ into a topological vector ...
1
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1answer
47 views

Topology and taxi cab metric

Let $A \subset \mathbb{R^k}$.Show that A is open if and only if it is open under the "taxi-cab metric" $d_{1}(x,y):=\sum_{j=1}^k|x_{j}-y{j}|. $ I was able to find that because $d_1(x,y)=\sum|x_j-y_j| ...
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3answers
44 views

3 questions about topology on metric space

I am reading a textbook about topology on metric space. I came over the following three 'Prove or Disprove' questions. Please: 1) comment on my work on the first two questions or leave me your own ...
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3answers
35 views

existence of unique fixed point

Let $(X,d)$ be a compact metric space and $f:X \to X$ satisfies $d(f(x), f(y))< d(x,y)$ for distinct $x$ and $y$. Then, show that $f$ has a unique fixed point. I tried this question by formulating ...
0
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1answer
18 views

A closed quotient map.

Let $X$ a compact, Hausdorff space and $R$ an equivalence relation on $X$. Prove that if $R$ is closed in $X\times X$ then the quotient map $q$ is a closed map. I like some advice. I have try ...
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1answer
22 views

Finding a special kind of continuous map on finite dimensional Hilbert Space

Let $H$ be a finite-dimensional Hilbert space, $B:=\{x∈H:∥x∥≤1\}$ be its unit ball Does there exist a continuous map $f:H→H$ such that $f(f(x))=x , ∀x∈H$, $f$ has no fixed points, and $f(B)$ is ...
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1answer
31 views

Borsuk - Ulam Theorem for $n=2$

Show that Borsuk -Ulam Theorem for $n=2$ is equivalent to the following statement : For any cover $A_1, A_2, $ and $A_3$ of $S^2$ with each $A_i$ closed, there is at least one $A_i$ containing a pair ...
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1answer
34 views

Is there a typo this theorem in Munkres's Topology? If not please explain

Doesn't he mean that the first inequality shows that $B_\rho \subset B_d$ instead? Since the distance $d(x,y)$ is larger than $\rho(x,y)$. The same goes for the second inequality.
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2answers
22 views

Show that the set $B := \left \{ (r,\infty) \mid r \in \mathbb{R}\right \} $ is a basis of some topology on $\mathbb{R}$, but not a topology itself.

Show $B := \left \{ (r,\infty) \mid r \in \mathbb{R}\right \} $ is a basis of some topology on $\mathbb{R}$, but not a topology itself. The definition of a topology basis is the following: A ...
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3answers
32 views

Set of limit points of topologist's sine curve $S$

Let $S=\{(x,\sin(1/x)):x \in (0,1]\}$ be the topologist's sine curve. Find the limit points $\lim S$ of $S$. I claim $\lim S = S \cup \{(0,y):y \in [-1,1]\}$. But, how do you show that any of these ...
2
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1answer
41 views

continuity and closure questions - topology

Let $(X,d)$ be a metric space. Let $U \subseteq (X,d)$. let $k \in (X,d)$. Prove that if $U$ is fixed, $d(U,k)$ is a continuous function of $k$. Prove that $\overline{U} = U \cup V$ where $V$ is the ...
0
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1answer
16 views

A surjective continuos mapping between $X$ and $Y$.

Let $p: X\rightarrow Y$ a continuos surjective and closed map such that $p^{-1}(y)$ is compact for every $y\in Y$. Prove that If $X$ is Hausdorff (regular) then $Y$ is Hausdorff (regular). Can ...
3
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1answer
45 views

Are any (non-empty) Euclidean open sets dense in the Zariski topology?

It's well known and easy to show that every Zariski open set is dense in the Zariski topology. However I search the web and didn't find an answer to my question, which I believe is true. My ...
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2answers
17 views

Let $(X,d)$ be a metric space. Let $U \subseteq (X,d)$. Let $k \in (X,d)$. Prove…

Let $(X,d)$ be a metric space. Let $U \subseteq (X,d)$. Let $k \in (X,d)$. Prove that $k \in U \Rightarrow d(U,k) = 0$, but the converse is not true. Prove that $d(U,k) = 0$ iff $k$ is a contact ...
0
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1answer
29 views

Two different topology in which $X$ is compact and Hausdorff. [on hold]

I' like a hint to solve the following problem Given $\tau, \tau'$ topologies to the set $X$. Prove that if $(X,\tau)$, $(X,\tau')$ are compact and Hausdorff then they are equal or are no ...
0
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0answers
19 views

Spectral Measure: Support well defined?

The support of a spectral measure is defined by: $$\mathrm{supp}E:=\bigcap_{C:E(C)=1}C$$ where $C$ are closed subsets (see german wiki). But why would then: $$E(\mathrm{supp}E)=1$$ Clearly, this ...
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0answers
23 views

What's the relationship between $\operatorname{bd}(A \cup B)$ and $\operatorname{bd}A \cup \operatorname{bd} B$?

$\renewcommand{\int}{\operatorname{int}}$ $\int(A \cup B)$ and $\int A \cup \int B$ $\operatorname{bd}(A \cup B)$ and $\operatorname{bd}A \cup \operatorname{bd} B$ $(A \cup B)'$ and $A' ...
0
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1answer
39 views

Prove that $\{x_{n}\} \subseteq A$ a Cauchy sequence $\Rightarrow$ $\{f(x_{n}\}\} \subseteq y$ is a Cauchy sequence.

Let $(Y,\rho)$ be a metric space and let $A \subseteq Y$ be a dense subset. Prove that $\{x_{n}\} \subseteq A$ a Cauchy sequence $\Rightarrow$ $\{f(x_{n}\}\} \subseteq y$ is a Cauchy sequence. Idea : ...
0
votes
1answer
27 views

Show $T_Q$ is a topology on $X$ [on hold]

Let $X$ be a set, $Q \subset X$. The $\textbf{characteristic topology}$ $$T_Q := \{ \emptyset\} \cup \{ U \in 2^X\mid Q \subset U\}$$ contains the empty set as well as all subsets of $X$ which contain ...
3
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1answer
22 views

Curves in $\mathbb R^n$ and boundary points

Thinking of a curve as a 1-dimensional continuum, intuitively one would think that a curve can never have this property. Perhaps intuition is correct and this question is somewhat naive, but here it ...
0
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4answers
41 views

How to show if a function f is open/not open

could you help me out? Let the real function $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)=x^2$. Show that $f$ is not open. How do I go about doing this? Thanks in advance!
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0answers
22 views

Trivial General Topology Question [on hold]

Consider the usual topology $\mathbb{U}$ on the real line $\mathbb{R}$. Determine whether or not each of the following subsets of $I=[0,1]$ are open relative to $I$ i.e. $\tau_I$-open. a) ...
0
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2answers
30 views

Prove the intersection of an ultrafilter is either empty or a singleton [on hold]

Let $F$ be an ultrafilter on a topological space $X$. Show that $\cap\lbrace $F$:$F$\in F\rbrace$ is either empty or a singleton set.
1
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1answer
20 views

Two intersecting disks with subspace topology in $\mathbb {R}^2$

Take two closed disks as subsets of $\Bbb R^2$ such that they intersect at exactly one point. Let $\Bbb R^2$ have the standard euclidean topology $\mathcal J_E$ and give the above set the subspace ...
0
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1answer
77 views

Topology and open sets problem

Let $C([a,b])$ be the set of continuous real valued function defined on the interval $[a,b]$, for ${-\infty<a<b<\infty}$. Define a subset $A \subset C([a,b])$ is open if, for every $f \in A$, ...
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2answers
36 views

Prove that int$(A) = \overline{[X-A]}$ where $[X - A]$ is the closure and the bar represents the complement.

Prove that int$(A) = \overline{[X-A]}$ where $[X - A]$ is the closure and the bar represents the complement. I have seen this proof before but I do like showing things like this via set theory. In ...
0
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0answers
34 views

What is the “distance from the nearest integer”? (is that a kind of norm?)

We define the function $$\Vert \cdot \Vert: \mathbb{R} \to \left[0,\frac 1 2\right]$$ as $$\Vert x \Vert = \min\big(x - \lfloor x\rfloor, 1 - (x - \lfloor x \rfloor)\big)$$ for $x \in \mathbb{R}$. I ...
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0answers
33 views

Is there any standard procedure to properly define a composite metric?

For example, space $A$ has a metric $\rho$, and its subspace $B\subset A$ has a metric $d$, which happens to have much better properties than $\rho$. So if $x_{1},x_{2}\in A\setminus B$, but they are ...
0
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1answer
25 views

Clarification on separability in Rudin

On pg 45 of Baby Rudin we have: 22. A metric space is called separable if it contains a countable dense subset. 24. Let $X$ be a metric space in which every infinite subset has a limit point. ...
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2answers
27 views

Algebraic (Kuratowski axiomatic) proof of a simple topological statement

I was trying to prove the following basic result using the Kuratowski closure axioms for topological spaces. Let $X$ be a space, $A$ and $U$ dense and open subsets respectively. Then $\overline{U} = ...
7
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2answers
131 views

Is bijection mapping connected sets to connected homeomorphism?

If $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$ is a bijection, mapping connected sets to connected, is $f$ necessarily a homeomorphism? The converse is true, a well known property of homeomorphisms. I ...
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4answers
54 views

How to give a rigorous proof of this fact about closures of open balls in the euclidean spaces?

Let $n$ be a positive integer, $\vec{a} \in \mathbb{R}^n$, and $r > 0$. Then it is intuitively clear that the closuer of the open ball $$B(\vec{a} ; r) \colon= \{ \vec{x} \in \mathbb{R}^n \colon ...
0
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0answers
32 views

Identifying Objects with Polygons

I can't seem to find anything regarding how one identifies something like a torus with am oriented square. I would like to know the significance of: How does the rectangle depict the torus? Why are ...
3
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1answer
35 views

Dense subsets and lipschitz functions

Let U and V be metric spaces. Let X be a dense subset of U. Suppose that there exists a function, h: X $\rightarrow$ V, such that h is Lipschitz with constant K. Show that there exists a function f: U ...
1
vote
1answer
29 views

Uniformly boundedness and equicontinuous set of functions

Let (X, $\mathcal{O}$) be a topological space. Suppose that {f$_{\alpha}$(x)}$_{\alpha \in A}$ is a family of functions on $\mathbb{R}$ that are uniformly bounded and equicontinuous. Prove that the ...