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
50 views

Is my approach right? Or is there a better way?

Let $E,F$ normed linear spaces, let $C$ connected of $E$, $D\subset F$, and $f:C\to D$ such that $f$ is open (i.e. sends open sets in $C$ "which is the same as open sets of $E$ intersected with ...
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0answers
15 views

Quickest way to restrict a homeomorphism

Let $\phi: U \to V \subset \mathbb{R}^n$ homeomorphism. My desire is: I want to say the restriction $\phi|_{\phi^{-1}(B_{r'}(x))}:\phi^{-1}(B_{r'}(x)) \to B_{r'}(x) $ is a homeomorphism in the ...
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2answers
60 views

Which of the following subsets of $\Bbb{R}^n$ is compact [closed]

Which of the following subsets of $\Bbb{R}^n$ is compact (w.r.t ususal topology of $\Bbb{R}^n$)- $\{(x_1,x_2, \dots x_n\ ): |x_i |< 1, 1\le i \le n\}$ $\{(x_1,x_2, \dots x_n\ ): x_1+x_2+ \dots ...
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0answers
16 views

Limits and Convergence of sequences in the form of $(k, k^2, 1/k)$

I'm dealing with proving the convergence and limits of sequences that are defined by multiple points, such as $$ \left(k, k^2, \frac{1}{k}\right) $$ and I'm not sure how to go about doing it. I'm ...
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1answer
41 views

$f$ is continuous $\iff f(\bar A) \subset \overline{f(A)}$

The problem is: $f:X\to Y$: any map. $f$ is continuous $\iff \forall A\subset X, \ f(\bar A) \subset \overline{f(A)}$ My understanding is: Suppose $f$ is continuous. $\forall A\subset X, A ...
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1answer
21 views

$X$ a topological space, what are the Borel sets of a closed subset $Y$?

Let $Y$ be a closed subset of $X$. Then $\mathcal B(X)$, the set of Borel sets of $X$, is the $\sigma$-algebra generated by all closed sets of $X$. Let $E$ be the intersection of $\mathcal B(X)$ ...
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2answers
58 views

$f: X \rightarrow Y$ is a continuous surjective function, $Y$ hausdorff and $X$ compact. proof that $f$ is open..

$f: X \rightarrow Y$ is a continuous surjective function, $Y$ hausdorff and $X$ compact. proof that $f$ is an open map.. "A function $f:X \rightarrow Y$ is an open map if whenever $U$ is an open ...
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2answers
279 views

Continuous surjections onto $\mathbb{R}$

I have two questions about continuous functions: Suppose $X \subseteq \mathbb{R}$ and $X$ has same cardinality as $\mathbb{R}$. Can we find a continuous function from $X$ onto $\mathbb{R}$? Suppose ...
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0answers
35 views

Prove that the property of being isolated is a topological property

Let $f:X \to Y$ be a homeomorphism and $A \subset X$ such that $A \cap A'= \emptyset$ prove that $f[A] \cap (f[A])'= \emptyset$ where $A'$ is the derived set of $A$ and $(f[A])'$ is the derived set ...
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0answers
22 views

Boundary of surface

Let $S$ be the region in $\mathbb{R}^2$ bounded by $x$-axis, $x=1$, and $y=x$. Define $$ f(x,y) = \begin{cases} 0 &\mbox{if } x = 0 \text{ or if $x$ or $y$ is irrational} \\ 1/q & \mbox{if ...
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1answer
63 views

Contractions - find $f$ such that $f:(0,1) \to (0,1)$ with no fixed points

The given solution I have is that $f(x) = 0.5 + 0.5x$ but I am not entirely how these results are derived. A function $f : X \to X $ is a contraction if $d(f(x),f(y)) \leq \gamma d(x,y)$ so here, ...
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7answers
674 views

How should one picture a topology/ topological space?

I can form a mental image of sets with structures like metrics or norms. But if I try to picture a topology/ topological space I fail every time. The information provided in Wikipedia confuses me ...
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2answers
65 views

Characterisation of limit points of subsets of Hausdorff spaces

The theorem which I want to show is the following: For a Hausdorff space $X$ and a subset $A$ of $X$, $x$ is a limit point of $A$ if and only if every neighborhood of $x$ contains infinitely many ...
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1answer
21 views

Convergence of the image of a sequence in topological sense

I haven't been able to find it, but i'm sure this question has been answered since it is a fundamemtal one: Let $(X, \tau $) and $(Y, \tau $) be topological spaces. Let $f: X\to Y $ be continuous. If ...
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1answer
46 views

Interior points of $A= \{(x,y): y=0 \}$

Find the interior points of $A= \{(x,y): y=0 \}$ in $\mathbb{R}^2$ I am working through some examples in preparation for an exam and I was fine with every example except the one given above. Can ...
0
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1answer
62 views

How can I prove that a set is connected?

For example,if $A \subseteq \mathbb{R}^2$ is finite, so $\mathbb{R}^2 \backslash A$ is connected. I'm trying to use the negation of the definition of connected metric space , so can I reach a ...
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2answers
61 views

example of a continuous function that is closed but not open

Give an example of a continuous function $f:\mathbb{R} \to \mathbb{R}$ that is closed but not open. $ f(x)=x^2$ is continuous and not open but It's not closed. What is an example? Thanks in advance.
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2answers
39 views

$C_c(X)$ is dense in $C_0(X)$

Let $X$ be a topological space, and let $C_0(X)$ be the $\mathbb{C}$-vector space of continuous functions $g:X \rightarrow \mathbb{C}$ with the property that for any $\epsilon > 0$, there exists a ...
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1answer
30 views

example of a open function such that the restriction is not open

Give an example of a function $f:X \to Y$ and a subset $A \subset X$ such that f is open but $f_A$, the restriction of $f$ to $A$ is not open. Can someone help me please? Thanks
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1answer
37 views

How to prove that $\Delta( A)$ with the Gelfand topology is compact and Hausdorff?

How do I prove: $\Delta( A)$ with the Gelfand topology is compact and Hausdorff. I've tried proving it closed, but I'm having difficulties with how to begin writing a proof. And I have no idea ...
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1answer
17 views

Induced Topology In Irreducible Subset

If T is a topological space, and X an irreducible subset of T, is it true that every open subset U of X, under the induced topology, is dense in X? I know the result is true for U open in T and ...
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1answer
32 views

Proof Question: Let $K\subseteq \mathbb{R}$: If any open cover for $K$ has a finite subcover, then $K$ is closed.

I have a question about the proof given by Stephen Abbott on theorem 3.3.8, specifically with this implication: Let $K\subseteq \mathbb{R}$: If any open cover for $K$ has a finite subcover, then $K$ ...
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0answers
20 views

Can this be a way to prove that the set of all limit points is closed

Let $x\in A'$ where $A'$ is the set of all limit points of $A$ . Let $x$ be a limit point of A'. This implies every neighborhood $x$ contains some p such that $p\neq x$ and $p\in A'$. This implies ...
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1answer
33 views

Topology: If $B\subset A$ and $f:A\to X$ is continuous, is $f|_{B}$ also continuous ?

Let $(A,\mathcal T_A)$ and $(B,\mathcal T_B)$ two topological space such that $B\subset A$ and $\mathcal T_B=\{B\cap U\mid U\in\mathcal T_A\}$. To me, if $f:A\to X$ is continuous refer to $\mathcal ...
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2answers
71 views

Next book in learning General Topology

I have just finished the book "C Adams & R Franzosa - Introduction to Topology. Pure and Applied". My aim is to reach to the level of the book "G E Bredon - Topology and Geometry". Bredon's book ...
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0answers
80 views

Fundamental group of the Klein bottle

Proof that fundamental group of the Klein bottle have the next two different presentations: a) $Gen(a,b \,; baba^{-1}=1)$, b) $Gen(a,b \,; a^{2}b^{2}=1)$. I have the proof of (a) and I can prove that ...
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2answers
95 views

Why the lens space L(2,1) is homeomorphic to $\mathbb{R}P^3$?

According to one definition of lens space $L(p,q)$, which is gluing two solid tori with a map $h:T^2_1 \rightarrow T^2_2$. And $h(m_1)=pl_2+qm_2$, $l_i$ means longitude and $m_i$ means meridian of the ...
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2answers
44 views

Find the limit points and exterior points of the following

Let $X=\mathbb R$, with the usual metric on $\mathbb R$ and $A=((0,1)\cap \mathbb Q)\cup$ {$2,3$}. Find the limit points of $A$, exterior points of $A$, $A^o$, $\overline A$ and $\partial A$. Can ...
3
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1answer
49 views

Connectedness of the Hausdorff distance.

Does anyone know a proof of connectedness of the Hausdorff distance? I mean a proof of the following: Theorem If $(X, \rho )$ is a connected metric space, then $(F(X), d_h )$ is also connected. ...
2
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1answer
22 views

Suppose $f\leq g,h$. If $\{ f<g\} \cap \{ f < h \} = \emptyset$, then $\{f<g \} \cap \overline{\{f < h\}} = \emptyset$.

Suppose $X$ is a Hausdorff topological space. Assume that $f,g:X \rightarrow \mathbb{R}$ are continuous functions such that $f \leq g$. Denote $\{f< g \}=\{ x \in X: f(x)<g(x) \}$. Lemma: ...
0
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1answer
33 views

Minimal uncountable well-ordered set $S_{\Omega}$ and the sequence lemma

Let $S_{\Omega}$ denote an uncountable well-ordered set every section of which is countable. In Munkres' book Topology, it says on pg 181 that: $S_{\Omega}$ satisfies the sequence lemma: if $A$ ...
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1answer
54 views

May two or more bases of different sizes generate a same topology?

I'm wondering the following example of the standard real topology: all open intervals with rational endpoints are a base for the standard real topology, as are the open intervals with irrational ...
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1answer
48 views

Continuous function on Polish space

I want to prove the following and have no idea how to proceed: For a continuous function $f: X \mapsto Y$ where $X$ is Polish and $Y$ is Hausdorff the following are equivalent: $f[X]$ is ...
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2answers
34 views

Continuous version of the Cantor-Schroder-Bernstein Theorem [duplicate]

Does the existence of continuous injections $f: A\rightarrow B$ and $g: B\rightarrow A$ imply the existence of a bicontinuous bijection between A and B (ie topological equivalence)? If not, what is a ...
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0answers
27 views

Continuos, surjective map $\pi$ is a quotient map $\iff$ $\pi$ sends saturated open to open or saturated closed to closed

Problem is: Continuos, surjective map $\pi$ is a quotient map $\iff$ $\pi$ sends saturated open to open or saturated closed to closed. ($U$ is saturated $\iff$ $\exists V \in Y$ s.t. $U = \pi^{-1} ...
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1answer
18 views

Minimal uncountable well-ordered set $S_{\Omega}$ is not compact

In Munkre's Topology, he writes that Minimal uncountable well-ordered set $S_{\Omega}$ is not compact, since it has no largest element. I know that $S_\Omega$ has no largest element, but how ...
2
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0answers
16 views

$U$ takes the same value on $\pi$ then $U$ is saturated

Let $\pi : X \to Y$ be any map, and $U$ be a subset of $X$. The problem is: "$\forall x\in U$, $ \pi (x) = \pi(x') $, then $x' \in U$" then $U$ is saturated. (U is saturated $\iff$ $\exists V ...
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3answers
263 views

reference for “compactness” coming from topology of convergence in measure

I have found this sentence in a paper of F. Delbaen and W. Schachermayer with the title: A compactness principle for bounded sequences of martingales with applications. (can be found here) On page 2, ...
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0answers
33 views

Help in Understanding the Proof of Baire-Category theorem

In the proof of the Baire category theorem(for non-empty Banach Spaces), I cannot understand the following Baire Category Theorem: A non-empty Banach Space cannot be a countable union of nowhere ...
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1answer
37 views

Mistake in the definitions of the linking number.

I am looking into the definition of the linking number. I've considered these two definitions. Consider a link $L$ with components $K_1$ and $K_2$, and respectively their embeddings $\gamma_1$ and ...
0
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1answer
36 views

products of uniform space

$X$ is a topological space(or even a nonempty set), and $(Y,Φ)$ is a uniform space, then $Y^X$ is a uniform space, too. $\widetilde{U}=\{(f,g):$for any $x\in X, (f(x),g(x))\in U\}$, ...
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1answer
73 views

Prob 12, Sec 26 in Munkres' TOPOLOGY, 2nd ed: Why we need continuity to show the result?

Let $f: X\mapsto Y$ be a closed continuous surjective map such that $f^{-1}(y)$ is compact, for each $y\in Y$. Show that if $Y$ is compact, then $X$ is compact. My question is why do we need $f$ to ...
27
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3answers
829 views

Why is the Hilbert Cube homogeneous?

The Hilbert Cube $H$ is defined to be $[0,1]^{\mathbb{N}}$, i.e., a countable product of unit intervals, topologized with the product topology. Now, I've read that the Hilbert Cube is homogeneous. ...
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1answer
37 views

Why do homeomorphisms tend to keep the shape?

We say that $\alpha:U \to V$ is a homeomorphism if $\alpha$ carries $U$ onto $V$ in a one-to-one fashion and if both $\alpha$ and $\alpha^{-1}$ are continuous. Now the word homeomorphism comes from ...
2
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1answer
41 views

Metric spaces as Cauchy complete categories, nlab entry, insight into a few of the constructions.

I'm having a bit of trouble making sense of some of the concepts in the "Metric space" section on nlab's entry on "Cauchy complete category" ...
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2answers
15 views

The Cauchy-Schwarz inequality for Hermitian forms

Let $V$ be a vector space over a field $\mathbb K$ and $f$ be a nonnegative Hermitian form on $V$. Then, $\forall x,y \in V$: $$|f(x,y)|^2 \le f(x,x)f(y,y)$$ Here's one proof: For an ...
1
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1answer
11 views

A sequence $(x_n)$ with infinite range $T$ converges if and only if $T$ has precisely one accumulation point.

Is the following statement true or false? A sequence $(x_n)$ with infinite range $T$ converges if and only if $T$ has precisely one accumulation point. I know that the statement is false, ...
2
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1answer
77 views

Prove: $m$ balls in $\mathbb{R^3}$ cut $\mathbb{R^3}$ into less than $m^3$ connected components.

I need to prove or at least to understand why $m$ balls in $\mathbb{R^3}$ cut $\mathbb{R^3}$ into less than $m^3$ connected components. But I've no idea how to deal with it. I even tried to draw it ...
20
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11answers
5k views

How to prove $[a,b]$ is compact?

Let $[a,b]\subseteq \mathbb R$. As we know, it is compact. This is a very important result. However, the proof for the result may be not familar to us. Here I want to collect the ways to prove $[a,b]$ ...
0
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2answers
39 views

Closed or open if it's continuous and not surjective?

If $f:[0,1]\rightarrow [a,b]$ is a continuous function, and $f([0,1])=(c,d)\subset [a,b]$. Is $f^{-1}([a,b])$ open or closed in $[0,1]$? If open: Since $[a,b]$ is closed so is $f^{-1}([a,b])$; ...