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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2answers
70 views

Why does one necessarily need the triangle inequality

I'm studying basic topological, metric and normed spaces and I am curious why one of the axioms of both a metric and a norm is the triangle inequality. It makes some sense to me having the triangle ...
1
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0answers
45 views

Three Jordan curves made from paths

For continuous paths $\phi,\psi:[0,1]\to \mathbb{R}^2$ such that $\phi(1)=\psi(0)$, let $\phi*\psi$ denote the composition path and $\phi^{-1}$ the inverse path. Consider points $p,q\in\mathbb{R}^2$ ...
1
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0answers
48 views

metric space with no perfect set

Let $X$ be a complete separable metric space containing no perfect set of size greater than $1$. In other words every subset of $X$ has an isolated point. It is well known that $X$ must be countable....
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0answers
35 views

Surjectivity on the image of a annulus

I'm trying to prove the Fundamental Theorem of Algebra as it is done in Birkhoff and MacLane. Unfortunately, I don't have access to the book, only to a sketch. Therefore, I'm filling the gaps myself. ...
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0answers
26 views

Minimal prime ideals of the ring of continuous functions

Let $X$ be a topological space. Are there any conditions on $X$ which guarantee that that the minimal prime ideals of $C(X)$, the ring of real-valued continuous functions on $X$, have a nice ...
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1answer
18 views

Union of Interiors is Subset of Interior of Union

I'm teaching my self topology using a book I found. This is the forth part of a 4 part question. links to other parts: one, two, three . I'm trying to prove the following problem from a book I found: ...
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2answers
39 views

Closure of Union contains Union of Closures

I'm teaching my self topology using a book I found. This is the second part of a 4 part question. First part is here. I'm trying to prove the following problem from a book I found: Let $X$ be a ...
10
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6answers
3k views

If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?

If a nonempty set of real numbers is open and closed, $\mathbb{R}$ Why/Why not? In other words, are $\emptyset$ and $\mathbb{R}$ the only open and closed sets in $\mathbb{R}$? Why/Why not? I tried ...
1
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0answers
17 views

Intersection of Interiors contains Interior of Intersection

I'm teaching my self topology using a book I found. This is the third part of a 4 part question. links to other parts: one, two. I'm trying to prove the following problem from a book I found: Let ...
0
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2answers
30 views

Continuous function with real numbers and rational numbers…

I know if I consider real numbers with topology which generated by [a,b) there is a continuous function from R onto rational numbers with usual topology. Also, I know there is a continuous function ...
0
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1answer
49 views

Connectedness of suspension of a topological space

The suspension $\Sigma X$ of a topological space $X$ is defined as the quotient space $$ \Sigma X=\dfrac {X\times \left[0,1\right]}{\sim}$$ where $(x,t)\sim (y,s)$ if and only if $s=t=0$, or $s=t=1$, ...
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0answers
27 views

Homotopy on a cylinder

Given a cylinder $C := \mathbb{R} \times S^1$, the fundamental group is $\pi_1 \cong \mathbb{Z}$. My basic question is: Why? I completely fail to see what the set of non-homotopic loops on the ...
1
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1answer
56 views

Is $Y$ homeomorphic to $\mathbb S^1$?

Let $Y = \mathbb S^1 \cup\{ (x,y): (x-2)^2 + y^2 =1 \}$ be a subspace of $\mathbb R^2$. Is $Y$ homeomorphic to $\mathbb S^1$? Is $Y$ homeomorphic to an interval? Can anybody please ...
0
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1answer
25 views

A bounded set in the plane

Consider the set in the plane described by the inequalities $x^2+3y\le e^y$, $y^2\le x+y$. Using Mathematica, I can see that the set is bounded. How can I prove that?
5
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4answers
3k views

Why is the empty set finite?

On page 25 of Principles of Mathematical Analysis (ed. 3) by Rudin, there is the definition (excluding the irrelevant parts for this question): Definition 2.4: For any positive integer $n$, let $...
6
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1answer
49 views

Example of set, finite outer measure, subsets, where outer measure does not converge

What is an example of a set $X$ and a finite outer measure $\mu^*$ on $X$, subsets $A_n \uparrow A$ of $X$, and subsets $B_n \downarrow B$ of $X$ such that $\mu^*(A_n)$ does not converge to $\mu^*(A)$ ...
14
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4answers
373 views

Why are continuous functions the “right” morphisms between topological spaces?

Recently, someone mentioned to me that given a function $f: X \to Y$ there are two natural functions between the powersets $P(X)$ and $P(Y)$. Namely $f: U \subset X \mapsto f(U)$ and $f^{-1}: V \...
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0answers
16 views

embedding $T_0$ topology with specific properties in $(\Bbb{R},τ')^J$

Let $X$ be an infinite set and let $τ$ be a $T_0$-topology on $X$ that has no open finite subset and has no finite closed subset except empty set. For $x∈\Bbb{R}$ let $U_x=${$y∈\Bbb{R}:y<x$}, and ...
5
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1answer
79 views

Is the sphere with a diameter homotopy equivalent to a surface?

This is for a homework problem: Take the unit sphere $\mathbb{S}^2$ and join the north and south poles with a line segment. Is the resulting space homotopy equivalent to a surface? Intuitively, ...
6
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5answers
702 views

Why can we always take the zero section of a vector bundle?

$\require{AMScd}$ As I understand it, a rank $k$ vector bundle is a pair of topological spaces with a map between them $$ E\xrightarrow{p}B $$ such that there exists an open cover $(U_\alpha)$ of $B$ ...
9
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1answer
160 views

Is a pathwise-continuous function continuous?

Suppose that $X$ is a locally connected and simply connected space and $f:X\to Y$ is a function such that for every path $\phi:[a,b]\to X$ the composition $f\circ\phi$ is continuous. Does it follow ...
1
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0answers
21 views

Motivation for the definition of continuous maps on topological spaces [duplicate]

In any category where the objects are sets equipped with certain relations and operations, the notion of "morphism" arises perfectly naturally. (Generally, a morphism between objects is one that ...
0
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1answer
39 views

The set of invertible $k \times k$ matrices with complex entries is a connected subset of $\Bbb C^{k \times k}$. [duplicate]

The set of invertible $k \times k$ matrices with complex entries is a connected subset of $\Bbb C^{k \times k}$. Required Hint for this problem. I have recently proved that the set of invertible $k \...
0
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1answer
40 views

Difference Symmetric of two closed sets are disconnected

Let $X$ be a topological space and let $A$ and $B$ be closed subsets of $X$.   Prove that if $A$ does not contain $B$ and $B$ does not contain $A$, then $(A\setminus B) \cup (B \setminus A)$ is ...
4
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2answers
51 views

Quotient of $\mathbb{R}^n$ with an unbounded equivalence class homeomorphic to $\mathbb{R}^n$?

Let $X=\mathbb{R}^n$. Suppose $X$ has an equivalence relation $\sim$ such that at least one class consists of a line (a $1$-D subspace) through the origin. If $X^*=X/{\sim}$, is it possible for $X$ to ...
3
votes
2answers
26 views

Restriction continuous function to be homeomorphism

Let $X$ be a compact space and $Y$ be a Hausdorff space and let $f$ be a continuous function from $X$ to $Y$. Let $$S:=\{y\in Y\mid \text{ the preimage of }y \text{ consists only one point}\}$$ ...
3
votes
1answer
47 views

is there coarsest $T_0$ topology on an arbitrary set?

Let $X$ be an infinite set and let $τ$ be a $T_0$-topology on $X$ witch is NOT $T_1$. Does $τ$ contain a $T_0$-topology on $X$ that is minimal with respect to $⊆$ ? Minimal $T_0$-topologies my ...
19
votes
2answers
741 views

Topological spaces in which every proper closed subset is compact

Let $X$ be a topological space. It is a basic result that that if $X$ is compact, then every proper closed subset $Y \subset X$ is compact. Out of curiosity, I would like to explore the converse of ...
1
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1answer
19 views

Find a subbasis $\mathcal S$ for $\mathcal T$ such that $\mathcal S$ does not contain any singleton sets.

Let $X$ be any infinite and $\mathcal T$ the discrete topology on $X$. Find a subbasis $\mathcal S$ for $\mathcal T$ such that $\mathcal S$ does not contain any singleton sets. I found the same ...
2
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2answers
53 views

Connectedness of punctured $G$

If I take a connected topological group $G$ and I remove the indentity from $G$, when I can say that $G-\lbrace 1 \rbrace $ is connected? Any suggestion or reference is appreciated.
2
votes
3answers
31 views

Slick proof that continuous image of precompact space is precompact

Is there a slick proof that precompactness is preserved under continuous images? By slick I mean analogous to the following proof that compactness is preserved under continuous images. Is it even ...
0
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2answers
23 views

Proving that only two points of intersection are sufficient for an accumulation point

My analysis textbook defines accumulation points as follows: Let $E$ be a set of real numbers. Any point $x$ (not necessarily in $E$) is an accumulation point of $E$ if for every $c>0$ the ...
-1
votes
1answer
61 views

Let $\mathbb S^1 = \{ (x,y) : x^2 + y^2 = 1 \}.$ Does either one of the following homeomorphisms hold?

Let $\mathbb S^1 = \{ (x,y) : x^2 + y^2 = 1 \}.$ Does either one of the following homeomorphisms hold? $\mathbb S^1 \cong(0,1)$ $\mathbb S \cong [0,1]$
0
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2answers
26 views

Sequence criterion for continuity

Let $f:X\rightarrow Y$ where $X$ and $Y$ are topological spaces. Is it true that $f$ is continuous at a point $c$ iff for every sequence $x_n$ converging to $c$, we have $f(x_n)$ converging to $f(c)$?
0
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1answer
21 views

Arcwise and pathwise connectivity in space filling curves

We know that a space filling curve is not injective from Netto's theorem. We know that a Peano space is a compact, connected, locally connected metric space. Essentially in pathwise connectivity ...
2
votes
1answer
28 views

Question about the proof of the fact that minimum is attained for a l.s.c. convex function over a convex compact set.

I quote here the proof of a result given in Haim Brezis Functional Analysis, Sobolev Spaces and partial differential Equations: I haven't been able to conclude where exactly is this hypothesis used: ...
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1answer
30 views

Separating two sets in normal topological spaces by binary relations

Trying to generalize Urysohn lemma or at least to rewrite its proof in a new way: Let $\mu$ be a normal ($T_4$) topological space on a set $\mho$. I will denote $\operatorname{up}\mu$ the set of all ...
1
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1answer
38 views

First Betti number definition

I found in the electric engineering literature this alternative definition of the first Betti number of an open set $\Omega\subset\mathbb{R}^3$ with Lipschitz boundary. $n_\Omega$ is the first Betti ...
4
votes
1answer
45 views

Possible typo: What does Bachman (Functional Analysis) mean with $\overline{A} = \overline{A}$?

From Bachman's functional analysis, here is theorem 7.1 Let $(X,\mathcal{O})$ be a topological space and let $A$ and $B$ be subsets of $X$. Then (1) $A\subset B \implies \overline{A} \subset \...
46
votes
12answers
4k views

What should be the intuition when working with compactness?

I have a question that may be regarded by many as duplicate since there's a similar one at MathOverflow. The point is that I think I'm not really getting the idea on compactness. I mean, in $\mathbb{R}...
1
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1answer
29 views

Solid tori, meridians, and longitudes

I am working through some of Rolfsen's "Knots and Links" and I have needed to go back and take a more careful look at the first few sections where he carefully discusses curves on solid tori. Let $V$ ...
1
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1answer
27 views

Connected CW-complex which is not locally finite

I am working on Topological Complexity of robot motion planning. I am looking for a connected CW-complex which is not metrizable. I have found that: Proposition 3.8. A connected CW complex $X$ is ...
1
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1answer
31 views

a problem from general topology

Let $X$ be a metric space with metric $d$,and fix a point $x_{0}\in X$. Let $\mathscr{T}$ be a topology on $X$ such that the function $f=$$d\left(x_{0},-\right):X\rightarrow \mathbb{R}$ is countinuous ...
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0answers
16 views

Fibers of unbounded linear functional are dense

I'm supposed to prove that if $f$ is a discontinuous linear functional $H\rightarrow \mathbb C$, each of its fibers $f^{-1} \left\{ \alpha \right\} $ is dense. I already know the kernel, i.e $f^{-1} \...
0
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0answers
29 views

What's the difference between the topology defined by a seminorm and the topology defined by the norm it induces?

I was just wondering whether there's some big difference between the topology generated by a seminorm and the norm it induces. For instance, Suppose $X$ is normed and $A$ is a subspace. $X/A$ is semi-...
0
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2answers
86 views

Prove for $x \in \mathbb{X}: x \in \bar{A} \Leftrightarrow d_A(x)=0$ [duplicate]

For $A \subset \mathbb{X}$ non empty and $x \in \mathbb{X}$ define the distance of x to A by $$d_A(x)=inf_{a \in A} d(x,a)$$ I am trying to prove for $$x \in \mathbb{X}: x \in \bar{A} \Leftrightarrow ...
1
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1answer
53 views

Find the limit point of set A in topology

Find limit points of set A A = {$n\sin\frac{1}{n}+(-1)^n\frac{1}{m}$ where $n,m\in N$} i think A' = {1, 0} but i can't prove this problem. Help
0
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1answer
31 views

Why is the empty set part of the topology in the following examples?

I am studying topology from Munkres. Why is the empty set an element of the topology that is schematically represented in the picture ?
0
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3answers
45 views

Show that the closure of $A$ is the intersection of all closed sets containing $A$, topology proof needed

I want to show that given $(X, \mathcal{T})$, we define $\overline A = \{x \in X| \forall U \in \mathcal{T}, x \in U \implies U \cap A \neq \varnothing\}$ (definition of closure from Munkres), then ...
5
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0answers
40 views

Does $\mu^*$ agree with $\mu$, measure space? [closed]

If $(X, \mathcal{A}, \mu)$ is a measure space, define$$\mu^*(A) = \inf\{\mu(B) : A \subset B,\,B \in \mathcal{A}\}$$for all subsets $A$ of $X$. I have a few questions? Is $\mu^*$ an outer measure? ...