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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Finite group products of finite groups have free subgroup of finite index

This is a problem in Hatcher's Algebraic topology. Show that a finite graph product of finite groups has a free subgroup of finite index, by constructing a finite-sheeted covering space of $K\Gamma$ ...
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7 views

Addition of continuous functions and compactness

Let $f$ and $g$ be continuous real or complex valued on locally compact space $ X$. If {$x \in X :|f(x)| \ge k $} and{$x \in X :|g(x)| \ge k $} are compact for every $ k > 0$, how can it be proved ...
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14 views

Is the space $B([a,b])$ separable?

Let $a$, $b$ be two real numbers such that $a < b$, and let $B([a,b])$ denote the metric space consisting of all (real or complex-valued) functions $x=x(t)$, $y=y(t)$ that are bounded on the closed ...
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10 views

Prove that a topological space $X$ is compact if and only if every centered family of closed subsets of $X$ has nonempty intersection.

Prove that a topological space $X$ is compact if and only if every centered family of closed subsets of $X$ has nonempty intersection. I have seen this idea a lot but everyone just assumes it ...
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1answer
41 views

What happens to the connectedness of $\mathbb R^2$ when countable many points are removed?

Does $\mathbb R^2$ remain connected when countably many points are removed? Does it remain path connected? This is not homework but is in response to working several problems where countable or ...
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25 views

Continuous function from standard topology to K-topology

I am just going through a practice problem and need to show that, where f(x)=x as the identity function. $f: \mathbb{R} \to \mathbb{R_{k}}$ where $\mathbb{R_{k}} = \{(a,b)- \{\frac{1}{n}\mid n \in ...
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16 views

$\text{span} \{ e_t \cdot w : t \in \mathbb{R} \}$ dense in $C_0(\mathbb{R}_+)$.

Let $\mathbb{R}_+ := [0,\infty )$ and let $w \in C_0(\mathbb{R}_+)$ be any function with $w(x) \neq 0$ for all $x \geq 0$. Why is $\text{span} \{ e_t \cdot w : t \in \mathbb{R} \}$, where $e_t(x) := ...
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15 views

$(X,\tau)$ needs to be $T_1$ in order to guarantee that $A'$ is closed?

Claim: If $(X,\tau)$ is not $T_1$ then $A'$ is not necessarily closed for any $A$. Proof: Here is an example. Take $X=\mathbb{N}$ and let $\tau=\lbrace u\subset\mathbb{N}\mid 1\in u\iff 2\in ...
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13 views

scott continuity, sub additivity

Let $(X, \sqsubseteq_x)$ and $(Y, \sqsubseteq_y)$ be two posets and let $\delta_x:X \to X$ and $\delta_y:Y \to Y$ be two closure operators (monotone, inflationary, idempotent). Then, a monotone ...
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34 views

Set of points of a sequence is closed

Let $x_n$ be a sequence of points in $\Bbb{R}$ such that $\displaystyle\lim_{n\to \infty}x_n=+\infty$. Why is $$X=\left \{ x_n\ : n\in \Bbb N \right \}$$ closed.($\Bbb{R}$ has the standard topology ...
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19 views

Prove continuous and Cauchy

Q1: Let (X,d1) and (Y,d2) be metric spaces and let T : X → Y be continuous function. Show that the image of an open set in X under T need not be open set in Y . Q2: If d1 and d2 are metrics on a set ...
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29 views

Prove it is Complete

Prove: Show that the subspace Y ⊆ C[a,b] consisting of all f ∈ C[a,b] such that f(a) = f(b) is complete. I stayed two hours to solve this problem can you help me
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26 views

Compact set example

Can you please give me an example of a set that is closed but not compact in R^2\Bbb? I know that a compact set is the one that is closed and bounded, and the set [a,b] is compact. But this question ...
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2answers
137 views

The real line, and finite complement topology is not a hausdorff

I am reading through Munkres pg 99 and it says without proof, "the real line in the finite complement topology is nota hausdorff space" i am having trouble getting started with proving this claim .. ...
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3answers
44 views

A few questions on homeomorphisms

I have been given the following questions on homeomorphisms, in particular showing the following spaces are not homeomorphic: $S^{1}$ and $[0,1]$, The letter 'X' and the letter 'Y', $S^{2}$ and the ...
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2answers
41 views

Show that the set $B:= \left \{ (a,b] \subset \mathbb{R} \mid a,b \in \mathbb{Q} \right \}$ is a basis…

Show that the set $B:= \left \{ (a,b] \subset \mathbb{R} \mid a,b \in \mathbb{Q} \right \}$ is a basis for some topology on $\mathbb{R}$, but that this topology is strictly coarser than the upper ...
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49 views

The set of rational numbers in the interval $(0,1)$ cannot be expressed as the intersection of a countable collection of open sets

The set of rational numbers in the interval $(0,1)$ cannot be expressed as the intersection of a countable collection of open sets I found this proof on a certain web page A direct proof would be ...
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1answer
20 views

Are non-empty perfect sets in separable metric spaces uncountable?

I know this is true for $R^{k}$, but can it be generalized to all separable metric spaces?
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33 views

Equivalence of categories $\Pi_1:\mathbf{K^1_{CW,*}}\to \mathbf{Grp}$

Let $K(G,1)$ denote the Eilenberg-Maclane spaces with fundamental groups isomorphic to $G$. Consider the category $\mathbf{K^1_{CW,*}}$ where objects are pointed $K(G,1)$ CW complex, and morphisms ...
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20 views

Characterization of Discrete Sets in R

Let A be a subset of $\Re$ . Does anyone have a characterization of discrete sets A ( which only have isolated points ) ? I'm coming up with A is discrete iff ( A is finite) or (A is infinite and ...
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5answers
57 views

Quick way to prove $\mathbb R^n-\{O\}$ is connected?

I want to show that the $n$-sphere is path connected, when $n>0$, and I've reduced it to showing/assuming that, if $n>1$, $\mathbb R^n-\{O\}$ is connected. Is there a quick way to prove this? I ...
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59 views

Reference Request to Prepare for Hatcher's “Algebraic Topology”

Hatcher himself has an excellent and always generously free set of notes on point- set topology: http://www.math.cornell.edu/~hatcher/Top/TopNotes.pdf It includes up to quotient spaces. It seems ...
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28 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 $X$ ...
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53 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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26 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
56 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 ...
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26 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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21 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.
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70 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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22 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 ...
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3answers
28 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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27 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 ...
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1answer
38 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
14 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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40 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
80 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, ...
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1answer
56 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 ...
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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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47 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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36 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 ...
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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
36 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
35 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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30 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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37 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 ...
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1answer
42 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 ...
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1answer
17 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 ...
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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 ...