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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11 views

For a Cantor set $\mathcal{C} \subset S^3$ such that $\pi_1(S^3 \setminus \mathcal{C})=0$, prove $S^3 \setminus \mathcal{C}$ can be split by a sphere.

I'm working from the paper Cantor Sets in $S^3$ with Simply Connected Complements by Richard Skora. On page 184 the second sentence states that any Cantor set $\mathcal{C} \subset S^3$ such that ...
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26 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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31 views

Question on $S_{3}/Z_{2}$ and $S_{2}/Z_{2}$ simple connectedness

I think is is clear to me that $S_{2}/Z_{2}$ is a hemisphere which is simply connected. How come $S_{3}/Z_{2}$ being disconnected? How can I understand this pictorially?
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57 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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1answer
32 views

Prove that square metric on $\mathbb{R}$ is in fact a metric.

In Munkres's topology, he proves that square metric on $\mathbb{R}^n$ is in fact a metric. By the square metric, I mean this function: $P:\mathbb{R}^n\times \mathbb{R}^n \rightarrow \mathbb{R}$ ...
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2answers
37 views

Why is the product of homeomorphisms a homeomorphism?

Let $f:A\to B$ and $g:A\to C$ be homeomorphisms, where $A,B,C$ are topological spaces. My book says that $(f\times g):A\to B\times C$ is also a homeomorphism. I wonder why that is. Define ...
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14 views

Easier proof of “countable hypocompactness”

I am interested in the following result, which appears as an old qual problem: Let $X$ be a metric space and $\{U_i\}$ a countable open cover. Prove that there exists a countable open refinement ...
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27 views

When is every open set a $F_\sigma$?

My question concerns the $F_\sigma$ property of a topological space $X$. I want to know if there is a particular name for those spaces in which every open set is $F_\sigma$. Moreover, if $X$ is a ...
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62 views

Show that $\overline{A} \cup \overline{B} = \overline{A \cup B}$.

Can someone please verify my proof? Show that $\overline{A} \cup \overline{B} = \overline{A \cup B}$. Clearly, $$A \cup B \subseteq \overline{A \cup B}$$ So, $$A \subseteq \overline{A \cup B}$$ ...
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1answer
40 views

Show that if $A$ is closed in $X$ and $B$ is closed in $Y$, then $A \times B$ is closed in $X \times Y$.

Can someone please verify my proof? Show that if $A$ is closed in $X$ and $B$ is closed in $Y$, then $A \times B$ is closed in $X \times Y$. Let $x \times y \in X \times Y - A \times B$. Then, ...
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2answers
79 views

Spaces where all singletons are closed

Do spaces where all singletons are closed have a name? I know for example that $\mathbb R$ is one of these spaces since the complement of a singleton $\{x\}$ is $(-\infty,x)\cup (x,\infty)$ which is ...
6
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1answer
174 views

Stone Cech compactification homeomorphism implies realcompactification homeomorphism

I was wondering: If $\beta X$ is homeomorphic to $\beta Y$, is it true that $\nu X$ is homeomorphic to $\nu Y$? Notation: If $f: X\rightarrow \mathbb R$, we denote it's extension by $f^\alpha: \beta ...
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20 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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1answer
26 views

zero-sets of $\beta X$

I'm trying to understand the following proof from Walker: Proposition. The zero-sets of $\beta X$ are countable intersections of closures in $\beta X$ of zero-sets of $X$. Proof. If $Z$ is a zero ...
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2answers
68 views

Mathematical description of bagel slicing into interlinked tori?

On his blog, MIT professor George W. Hart (http://www.georgehart.com/bagel/bagel.html) depicts decomposing a bagel, i.e., solid torus, into a pair of interlinked solid tori. I presume this is a known ...
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39 views

classifying topological spaces by measures

While looking at some spaces, I happend to know,that in some spaces(like $\mathbb R^n$) Null sets have topological properties(defining the Algebra by the open sets)! some examples: in $\mathbb R^n$ a ...
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1answer
29 views

Homogeneous topological space with the fixed-point property

Let $X$ be a topological space. $X$ is said to be homogeneous if for every $x$ and $y\in X$ there is an self-homeomorphism $f$ of $X$ such that $f(x)=y$. Further, $X$ is said to have the fixed-point ...
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1answer
37 views

Deck transformations

We have a theorem that says that if a group $G$ acts on a path-connected space $Y$ properly discontinuously, then $\pi: Y \rightarrow Y/G$ is a covering map. Especially, $G$ is isomorphic to the group ...
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2answers
44 views

Examples of continuous non-transitive group actions

In studying topology, I encountered this problem: Let $S$ be a topological space and let $G$ be a topological group acting continuously on $S$ (group action as $G \times S \to S$ map is continuous). ...
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0answers
31 views

When is an outer Borel regular measure (inner and outer) regular?

Let $X$ be a topological space and $\mu$ an "outer" Borel regular measure on $X$ (for all $A\subset X$, there is $B$ Borel with $\mu(A)=\mu(B)$). Assume that $X=\cup _{i=1}^\infty U_i$, where each ...
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3answers
92 views

Is the function $f(x) = 1/x$ continuous?

A function f is mapped from the non-zero reals to the reals . We assume the natural topology to be induced on the domain. Then is the function f(x) = 1/x continuous ? EDIT Suppose I use this ...
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0answers
23 views

Radial Division of a Figure into Equal Parts

Given an $n$-gon, $P$, for which numbers $k$ must there exist a point $x$ so that there are $k$ equally spaced rays emanating from $x$ which divide $P$ into $k$ equal area parts? For $k=2$ we can ...
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2answers
26 views

Show that the component in $X$ of $a$, $C_X(a)$, is closed.

Show that the component in $X$ of $a$, $C_X(a)$, is closed. The definition I know is $C_X(a)$ consist of all connected set, which consist of $a$. I have no idea on how to utilize the "clopen set ...
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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 ...
4
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2answers
41 views

Finding points where a smooth map between differential manifolds is or is not an immersion.

I am having trouble answering questions pertaining to immersions on smooth manifolds. For example: Given the unit sphere $S^2$ around the origin in $R^3$ and the map $f: S^2 \rightarrow R^3$ given ...
5
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2answers
83 views

Prove that $\mathbb{R}^{n}-A$ with the standard topology is connected where $n \geq 2$ and $A \subset \mathbb{R}^{n}$ is countable.

I've been stuck on this proof for quite a while. While I realize it is much easier to show using arcwise connectedness or pathwise connectedness, I would like to complete the proof without resorting ...
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0answers
44 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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40 views

Baker's transformation: continuity, orbits of irrational and rational points

I've reading the Pugh's Analysis book and I have problems with one exercise. This says: The baker's transformation: a rectangle of dough is stretched to twice its length and folded back on itself. ...
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2answers
69 views

Prove that a function of the rational numbers $\mathbb{Q}$ with subspace topology inherited from $\mathbb{R}$ is injective

Consider the set of rational numbers $\mathbb{Q}$ equipped with the subspace topology inherited from $\mathbb{R}$. Suppose $g: \mathbb{R} \to \mathbb{R}$ and $h: \mathbb{R} \to \mathbb{R}$ are ...
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2answers
65 views

If a function $f:\mathbb R\to\mathbb R$ is proper, then it tends to infinity as $x\to \infty$

I came across this problem the other day, I've played around with it but still don't really have any ideas: "A function, $f$, is said to be proper if the pre-image of any compact set is compact. Show ...
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1answer
53 views

Characterization of dense open subsets of the real numbers

Does the complement of every dense open subset of the real numbers have Lebesgue measure $0$? This is certainly not a characterization of dense open subsets of reals, since the complement of the ...
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1answer
45 views

Is $h$ and homeomorphis of $\mathbb{R}^{\mathbb{N}}$ on itself?

I have the following problem: Given the sequences $(a_1, a_2, ...)$ and $(b_1,b_2,...)$ of real numbers with $a_i>0$ for all $i$, define $h:\mathbb{R}^{\mathbb{N}}\longrightarrow ...
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1answer
39 views

What does it mean that the set of polynomials is dense in $C^0([a,b],R)$

What does it mean that the set of polynomials is dense in $C^0([a,b],R)$ $C^0( [a,b ], R )$ is the set of continuos functions. As I understand it, for the set of polynomials (call this set $P $) to ...
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2answers
317 views

Is this modified coffee cup equivalent to some n-fold torus?

The familiar joke is that a coffee mug is topologically equivalent to a donut. I own a coffee mug that is essentially a regular coffee mug with a tube going the middle. I'm not sure how to describe ...
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1answer
39 views

Is there a name for a property defined in terms of open sets?

We know that if a property is defined in terms of open sets then the property is preserved under a homeomorphism. Is there a name for such a property?
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2answers
35 views

A Hausdorff space which is not completely regular

My example is, $f : \mathbb{R}^+ \to \mathbb{R}$ defined by: $$f(x) = \begin{cases} x, &\text{if }0 \leq x < 1 \\ \tfrac{1}{x}, &\text{if }x \geq 1. \end{cases}$$ Even though $f(0)=0$ but ...
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3answers
40 views

Show that $S=\mathbb R^2\setminus\{(a,b):a,b\in\mathbb Q\}$ is path connected. [duplicate]

Show that $S=\mathbb R^2\setminus\{(a,b):a,b\in\mathbb Q\}$ is path connected. By definition of path connected, there should exist continuous mapping $f:[0,1]\rightarrow \mathbb R^2$ s.t. ...
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2answers
28 views

Showing that two maps are homotopic

Let $X$ be a topological space and let $S^2 \subset \mathbb{R^3}$ be the unit sphere with the metric $d$ inherited from $\mathbb{R^3}$. Show that if $f,g:X\to S^2$ are continuous maps such that ...
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1answer
43 views

Prove that a set is compact

Let $X$ be a compact space, let $U$ be an open set in $X$, Let $f:U\to [0,1]$ be a continuous map. Prove that the set $$K=\{(x,t): x \in U , 0 \leq t \leq f(x) \} \subset X \times [0,1]$$ is compact. ...
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43 views

Proving a set is dense in R if all limit points are in R

Prove: $E$ is dense in $\Bbb R$ if and only if the set of limit points of $E$ equals $\Bbb R$. That is, $$E′=\{x\mid \hbox{$x$ is a limit point of $E$}\}=\Bbb R\ .$$
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1answer
45 views

True or False: Topological Group and $S^1 \vee S^1$

$i.$ $S^1 \vee S^1$ can be embedded in a topological group $ii.$ $S^1 \vee S^1$ can be covered by a topological group I think $i.$ is true since we can embed the wedge sum into $\mathbb{R}^2$, which ...
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0answers
42 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 ...
4
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3answers
62 views

Relation between continuous maps and convergence of sequences

I am studying metric spaces and I know that in a normed space $E$ a map $T:E \to E$ is contínuous if and only if $T(x_n) \to T(x)$ for every convergent sequence $x_n \to x$ in $E$. In my notes there ...
0
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1answer
46 views

Is there a discrete initial topology on the set of real numbers?

Consider the real numbers R first as just a set with no structure. Then consider it as a topological space R* with the usual topology. The question is: is there a function f from R to R* whose ...
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32 views

Characterising subgroup

Let $\omega $ be a path in $\hat{X}$ with $\omega(0), \omega(1) \in p^{-1}(x_0)$, where $p$ is a covering map $p:\hat{X} \rightarrow X$. Let $\alpha=[p \circ \omega] \in \pi_1(X,x_0)$. Then we have ...
4
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3answers
38 views

Does continuity of $f$ imply $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$?

I'm struggling to prove or disprove that the continuity of $f$ implies $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$. $f:X\to Y$ is a map between metric spaces $(X,d),(Y,d')$ while $\bar M$ denotes the ...
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2answers
57 views

On invertible matrix in $\mathbb R^{n^2}$ [on hold]

How do i prove that the invertible matrix form an open and disconnected set in $\mathbb R^{n^2}$ or generally if $G$ its a multiplicative group of matrices in $\mathbb R^{n^2}$ with Int($G$) non ...
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1answer
17 views

Constructing semi-regular spaces

This is essentially a question from Engelking's text. Suppose that $( X , \mathcal{O} )$ is a (Hausdorff) space, and let $\mathcal{O}^\prime$ be the topology on $X$ generated by the family of all ...
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0answers
34 views

At most one connected component of unbounded portion of entire function.

Suppose $f$ is an entire complex analytic function and $R$ a positive real number. Define the set $E:= \{z\in\mathbb{C};|f(z)| < R\}$ to be the set of $z$ whose image is bounded by $R$. I want to ...
6
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1answer
62 views

Show that there is sequence of homeomorphism polynomials on [0,1] that converge uniformly to homeomorphism

Let $f:[0,1]\rightarrow [0,1]$ be a homeomorphism. Show that , there exists a sequence of polynomials $$(P_n(x))_n$$ such that $P_n(x)$ converge uniformly to $f$ on $[0,1]$ and every $P_n(x)$ is a ...