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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Open set in subspace not open in the entire space example

I am stuck with the following problem: X is a metric space. Suppose that Y is a subspace of X. Give an example that an open set in Y is not open in X. My own approach was this: Suppose U is a subset ...
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50 views

Uniqueness of a continuous extension of a continuous map from a set to its closure

Suppose $f$ is a continuous map from a space $A$ to a Hausdorff space Y. Then I know that $f$ can be extended uniquely to a continuous map from closure of $A$ to Y. What is a counterexample to the ...
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97 views

A perfect Hausdorff space that is not metrizable.

Can anyone provide an example of a well-known topological space that has the following three properties: (1) It is perfect (contains no isolated points), (2) T2, and (3) not metrizable.
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142 views

Good book on Probability theory, Topology and Group theory for a beginner.

This year I will start three new 'branch' in mathematics : Probability theory, topology and group theory. I would like to know three complete books i.e. starting with the basics 'tools' whilst going ...
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81 views

Basis For A Topology

Let $X$ be a non empty set and $A$ is a subset of $X$. Show that the family of all subsets of $X$ which contains $A$, together with the empty set, forms a topology on $X$. (Use definition of basis ...
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107 views

Is the suspension of a countable collection of points in $\mathbb{R}$ a countable collection of circles?

I am extremely new to topology and taking an algebraic topology course, and I need some help understanding the behavior of suspensions. The problem I am working on asks about the suspension of the ...
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83 views

Is a partially topological group completely regular

Let $G$ be a group and $\mathcal T$ be a topology on $G$ and the function $$ \begin{align*} &f:G\times G\to G\\ &f(x,y)=xy^{-1} \end{align*} $$ be continuous at $(1,1)$. Is $(G,\mathcal T)$ ...
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38 views

Paradox in connection with definition of limit points and order limit theorem?

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I come across something that appears (to me) as a paradox. Let me first write down one definition and two theorems that ...
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50 views

Connectedness and non-local-connectedness of a subspace of $\mathbb R^2$

Let $(X,\tau)$ be the subspace of $\mathbb R^2$ consisting of the points in the line segments joining $(0,1)$ to $(0,0)$ and to all the points $(1/n,0)$, $n=1,2,3,\ldots$. Show that $(X,\tau)$ ...
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124 views

A continuous bijection from a Hausdorff space to a non-compact space which is not a homeomorphism

Recall the following theorem: Let $X$ be a compact space and $Y$ a Hausdorff space. Suppose that $f:X \rightarrow Y$ is a continuous bijection. Then f is homeomorphism. Prove that the ...
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73 views

A continous map between the two torus and the torus

Let $\Sigma$ be the doubled torus (a compact oriented) surface of genus 2) and let $T$ be the torus. Suppose $f: \Sigma \rightarrow T$. Prove that $f$ is not a local homeomorphism. Attempt at ...
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37 views

Show that $K:=\left\{z\in\mathbb{C}: \lvert z\rvert =1\right\}$ is a topological group.

As the title already says I have to show that $$ K:=\left\{z\in\mathbb{C}: \lvert z\rvert =1\right\} $$ is a topological group. First of all, $K$ is a group concerning the ...
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82 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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1answer
90 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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53 views

If $p:E \to B$ is a covering map, and if $E$ is compact, prove that $p^{-1}(b) $ is finite for all $b \in B$.

If $p:E \to B$ is a covering map, and if $E$ is compact, prove that $p^{-1}(b) $ is finite for all $b \in B$. I need to verify correctness of my proof and ask if there is a more straight-forward ...
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52 views

A trouble with the discrete product topology

Consider a finite set $S=\{1,\ldots,n\}$ with the discrete topology, and moreover construct the product topological space $S^\mathbb N$ with the product topology. $S^\mathbb N$ is made by all the ...
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78 views

Dense subsets of $(L^p(\Omega),\|\cdot\|_p)$

The following results hold. Theorem Let $\Omega\subset\mathbb{R}^n$ be an open set. Then $C^0_c(\Omega)$ is dense in $(L^p(\Omega),\|\cdot\|_p)$, if $1\le p<\infty$. Theorem Let ...
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39 views

What is the line going through points $(5, 5, 5), (2, 2, 2) \in \mathbb{R^3}$ when mapped it is mapped to a point in the real projective plane?

So the real projective plane is homeomorphic under a function $f$ to $\mathbb{R^3} - (0, 0 ,0)$. Hence lines in $\mathbb{R^3} - (0, 0 ,0)$ become points in the the real projective plane. So what is ...
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1answer
40 views

Any polynomial function is continuous - what about a constant function?

I read that any polynomial function is continuous. I.e. If we have an open set $U$ in the range, $f^{-1}(U)$ will be open in the domain. Let $\mathbb{R}$ have the standard topology. Define $f: ...
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74 views

Sequentially compact space

Is every sequentially compact space metrisable? If not, then, can you give me an example of a sequentially compact space that is not compact.
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92 views

A question about open balls in Hilbert space.

Let $S$ be a finite dimensional Euclidean space and let $B$ be an open ball of $S$. If $f$ is any homeomorphism of $S$ onto itself, then (it is easy to see that) $f(B)$ is a bounded and connected open ...
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175 views

sequential continuity vs. continuity

A short and hopefully simple question for someone with more experience in topology: If a topology is induced by a mode of convergence and in fact nothing more is known apriori, wether if this topology ...
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77 views

Some questions about the proof of the General Linear Group being a manifold.

I understand the idea behind proving that GL(n,$\mathbb{R}$) is a smooth manifold by first using the fact that it is isomorphic to $\mathbb{R}^{n^{2}}$ and using the continuity of the determinant ...
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47 views

What is the subject to study continuous function?

What is the subject specialized to study continuous function? I am not sure if it is the topology and what topics of topology study continuous function?
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121 views

With which separation axiom is the quotient space well-behaved?

On the wikipedia page about quotient spaces one can read the following: In general, quotient spaces are ill-behaved with respect to separation axioms. The separation properties of $X$ need not be ...
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1answer
38 views

Is the set discrete?

Is the set $S=\{(m+\frac{1}{2^{|p|}},n+\frac{1}{2^{|q|}}):m,n,p,q\in \mathbb{Z}$} discrete in $\mathbb{R}^2$? I'm not getting how shall I check discrete here?
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80 views

Quotient of a locally compact space

I am looking for an example of a quotient of a locally compact space that isn't locally compact. Is there a not too complicated example ?
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55 views

$\mathbb{R}\mathbb{P^2}$ as a quotient space

If we construct $\mathbb{R}\mathbb{P^2}$ by gluing the sides of $I^2$, can anyone explain for me that why $\mathbb{R}\mathbb{P^2}$ can be considered as the quotient space induced by the equivalence ...
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66 views

Does there exist a Lipschitz map from the unit interval onto the unit square?

It is well-known that continuous space-filling curves exist. But can they be Lipschitz? Specifically, is there a Lipschitz map from [0,1] onto [0,1]x[0,1]?
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38 views

Is $\textrm{im}(f)$ homeomorphic to the torus less the inner equator?

Consider the map $id_{S^1}\times f:S^1\times [0, 1]\longrightarrow S^1\times S^1$ where $f:[0, 1]\longrightarrow S^1$ is given by $$f(t)=(\cos(\pi t), \sin(\pi t)).$$ Is it true that $\textrm{im}(f)$ ...
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60 views

Residually Finite group $\Rightarrow$ Totally disconnected

How can I prove that a residually finite group $G$ is totally disconnected? I considered the topology generatad by $\{Ng\}_{N\in\eta,\;g\in G}$ where $\eta=\{N\unlhd G \;, |G:N|<\infty\}$ and I ...
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72 views

If $Y$ is compact and $f : X \rightarrow Y$ is a map whose graph $G = \{ (x,f(x) : x \in X\}$ is closed in $X \times Y$ , then $f$ is continuous.

If $Y$ is compact and $f : X \rightarrow Y$ is a map whose graph $G = \{ (x,f(x) : x \in X\}$ is closed in $X \times Y$ , then $f$ is continuous. Let $C \subseteq Y$ be a closed. Let $x \in X - ...
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56 views

Determining if certian properties of a topological space pass to its image under a quotient map.

A property $P$ of topological spaces is said to "pass to quotients" if whenever $p : X \rightarrow Y$ is a quotient map and $X$ has property $P$ then $Y$ has property $P$. For the following ...
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1answer
120 views

Sphere homeomorphic to plane?

I just took a course in general topology about a month back, and I was wondering whether it was possible to explain why the Earth seems flat from our point of view but is in fact a sphere using the ...
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79 views

The mathematics of anyons

I've recently learnt about particles called anyons which exist within a two dimensional framework. Which I find quite strange since we, well live in a three dimensional world. I've also found out that ...
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76 views

How can I show that the closure of the set $\Phi = \{1, 2, 3, 4,\ldots\}$ is the set itself?

If $\Phi \subseteq C(I)$, where $C(I)$ is the set of continuous real-valued functions on the interval $I=[0,1]$, and we define $\Phi = \{\phi_1, \phi_2,\phi_3,\ldots\}$ where $1=\phi_1$, $2=\phi_2$, ...
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1answer
76 views

Show that topologies are the same

I just read a proof where it was said that if for each element in the topology 2 we find an element in topology 1 that is contained in this set and vice versa, then they are the same. How do I see ...
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135 views

Constructing a sequence that is pointwise bounded but not uniformly bounded by points in a closed, nowhere dense set in $\mathbb{R}$.

I believe that this question below is asking for a sequence of functions that are bounded pointwise in $\mathbb{R}$ but NOT uniformly bounded in a closed, nowhere dense set of $\mathbb{R}$. Suppose ...
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99 views

Assume that $(\text{X}, T)$ is compact and Hausdorff. Prove that a comparable but different topological space $(\text{X},T')$ is not.

Say that a topological space is CH if it is both compact and Hausdorff. Let $T$ and $T'$ be two topologies on the same set X that are comparable but different, i.e., $T$ is either strictly ...
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40 views

Understanding on the basis of topology on infinite product space

For infinite product space, the product topology is generated by the open sets that place open restrictions only in finitely many coordinates. I try to understand why it must be the restrictions on ...
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55 views

Is the complement of an open ball in a Banach space connected?

Let $B$ be a real Banach Space whose dimension is at least $2$, and let $S$ be a subset of $B$ that is an open ball. Is the complement of $S$ (with respect to $B$) always connected? Idea One could ...
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141 views

Every closed subset $E\subseteq \mathbb{R}^n$ is the zero point set of a smooth function

In Walter Rudin's Principles of mathematical analysis Exercise 5.21, it is proved that for any closed subset $E\subseteq \mathbb{R}$, there exists a smooth function $f$ on $\mathbb{R}$ such that ...
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120 views

Show that two discrete spaces are homeomorphic iff they have the same cardinality

I have the following question: Show that two discrete spaces are homeomorphic iff they have the same cardinality: I have tried the following: Let $f: (X, \mathcal{T}_{discrete}) \to (Y, ...
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55 views

In the proof of every isomorphism of $\mathbb{C}^n$ onto an $n$-dimensional subspace of a complex topological vector space is a homeomorphism

I was reading the proof of the following theorem in Rudin 2/e: Theorem 1.21 If $n$ is a positive integer and $Y$ is an $n$-dimensional subspace of a complex topological vector space $X$, then ...
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60 views

Compactness, topology

In a general topological space $(X,\tau)$ I have the following situation: $$F\subset M\subset N$$. If I prove that $F$ is compact in $N$ (w.r.t the induced topology), is it true that $F$ is compact ...
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51 views

Is image of every closed set is closed set ?

If f is continuous and bijection function of Hausdorff space into topological space. Is image of every closed set is closed set ? Thanks.
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68 views

Complement of open unit disk homeomorphic to $\mathbb{R}^2$

Is $\{(x,y) \in \mathbb{R}^2 | x^2+y^2 \geq 1\}$ homeomorphic to $\mathbb{R}^2$? I suppose that they are not, because they have different fundamental groups.
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88 views

Elementary topology problem

A function $f: \mathbb{R} \to \mathbb{R}$ is said to be bounded at a point $x_0$ provided that there are positive numbers $\varepsilon$ and $M$ so that $|f (x)| < M$ for all $x \in (x_0 - ...
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51 views

On Path-Connectedness

Prove that if there is $x_0 \in X $, such that for any $x \in X$, there is a path connecting $x$ to $x_0$, then $X$ is path connected. I have trouble in provving it. Here is my attempt: Fix a $x,y ...
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44 views

If $X, Y$ have the included point topology, $f: X \to Y$ is continuous iff $f$ preserves the included points

For a prep exam: My question is the following: If $X, Y$ have the included point topology, $f: X \to Y$ is continuous iff $f$ preserves the included points. Here is the definition of included point ...