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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3
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1answer
43 views

interior of union of two sets

I have been reading about some properties of interior and closure operator. I came across the fact that For any topological space $X$ and $A$ and $B$ $\subseteq X$.It is not true in general $i(A \cup ...
5
votes
1answer
49 views

Every metric space is a D-space.

I think it is correct, but I would like another pair of eyes to verify. Definition. An open neighborhood assignment is a function $f:X\to \tau$ such that $x\in f(x)$. Definition. A space is said to ...
0
votes
5answers
62 views

Open cover for $(0, 1)$

Would the set $(-2, 3)$ be an open cover for $(0, 1)$? In that case wouldn't $(-1, 2)$ be a subcover for $(0, 1)$? There's only 1 set in the subcover collection, thus the collection is finite. ...
3
votes
3answers
528 views

Any finite set is compact; what exactly is a finite set?

Any compact set is finite. Assume the sets are in $\mathbb{R}$ Since $A = [0, 1]$ is compact, it is also finite. As for $B = (0, 1)$, it is not compact, so it is infinite. However, how is it infinite? ...
1
vote
1answer
35 views

Is function $f$ also uniformly continuous?

I've been thinking on the following problem lately: Let $(X,d)$ be a metric space and $f_1,f_2,...,f_n: X \rightarrow \mathbb{R}$ and $f(x) = \max\{f_1(x),f_2(x),...,f_n(x) \}$,$x\in X$ If the ...
5
votes
1answer
70 views

Topology generated by Functions.

Fix a set $X$. Let $\mathcal{F}_0$ be a set of functions $g:X\to\mathbb{R}$. Let $\mathcal{T}_0$ be the smallest topology on $X$ in which all $g\in\mathcal{F}_0$ are continuous. Next, we say that a ...
1
vote
1answer
26 views

Any Typo in This Open Set & Closure Operation Problem?

I am working on a problem from a class-note that is rife with typos, and I think there is also a bug here in this problem for I could not make any sense out of it: If $\tau $ is a set of open sets in ...
-7
votes
1answer
42 views

What can we say about the set $\cap U_i$? [closed]

Let $U_1\supset U_2 \supset......$ be a decresing sequence of open sets in Euclidian 3 space $\mathbb R^3.$ What can we say about the set $\cap U_i$? A. It is infinite. B. It is ...
7
votes
1answer
19 views

nonempty open set in normed space is connected iff each pair of points of the set can be joined by a polygon that lies wholly in the set

Let $E$ be a normed vector space. Let $x_1, \dots, x_m$ be points of $E$. Let $f(t) = (k-t)x_k + (t - k + 1) x_{k+1}$ for $k-1 \le t \le k$, $k = 1, 2, \dots, m-1$. The set $\{f(t)\text{ }|\text{ }0 ...
2
votes
0answers
52 views

covering spaces

A covering space of a Hausdorff space is also Hausdorff. Conversely, a compact Hausdorff finite covering space has a Hausdorff base space. However, in general, a non Hausdorff space may have a ...
2
votes
0answers
37 views

Van Kampen Theorem for a Certain Square

Take a square with all the edges identifies. Choose a point $x$ on the boundary of this square. Take a small neighborhood $B_\epsilon(x)$ of this point. I want to compute $\pi_1(B_\epsilon(x) ...
0
votes
2answers
35 views

An other definition of continuity

I know that $$f: E\rightarrow F~\text{is continuous}~\Longleftrightarrow \forall V ~\text{open in}~F, f^{-1}(V)~\text{is open in}~ E$$ How to prove that $$f: E\rightarrow F~\text{is ...
0
votes
0answers
30 views

The “retraction” of $S\subseteq\Bbb R^2$ has rectifiable boundary

This is a continuation of the line of investigation of What's the most efficient way to mow a lawn? (although this question is self-contained). For $S\subseteq\Bbb R^2$ and $x\in\Bbb R$, define ...
0
votes
2answers
23 views

Wedge Sum Embedding with Inclusions

Let $X$ and $Y$ be two disjoint topological spaces, $x_0\in X$, $y_0\in Y$ and we consider the Wedge Sum (the quotient of the union by the relation $x_0\sim y_0$). I want to proof that $\pi \circ ...
0
votes
3answers
64 views

How to show that a real continous function with image in the rationals is constant?

Can someone please explain to me how I am supposed to approach this question: If $f: [0,1] \to \mathbb{ R}$ is continuous, and has only rational values, then $f$ must be a constant.
1
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0answers
42 views

If a set is closed, why is that set intersected with a compact set closed?

If F is a closed subset of K and K is compact, why is F intersect K closed?
2
votes
1answer
22 views

Continuity and the closure

I want to prove that $$f:E\rightarrow F~\text{is continuous}\Rightarrow \overline{f^{-1}(B)}\subset f^{-1}(\overline{B}),\forall B\subset F$$ I say let $B\subset F$ and let $x\in ...
2
votes
1answer
18 views

A question regarding Surface Integrals and Stoke's Theorem

Let $G$ be an open set in $ \Bbb R^3$ and $F:G \rightarrow \Bbb R^3-{0}$ a vectorial field of class $C^1$. Suppose that $S$ is an open set, contained in $G$, whose non-empty boundary $\delta S$, is ...
2
votes
1answer
25 views

Terminology in Viro et al.

I'm working through this book (Elementary Topology) and skimming the first section to make sure I'm not missing anything important to begin an independent study in algebraic topology, and I've come ...
4
votes
0answers
18 views

Viewing Homotopies as Paths in $\mathcal{C}^0(X,Y)$

When I think intuitively about homotopies, I think about them as paths between two functions. This is more comfortable and suggestive than any categorical talk about "morphisms between morphisms", so ...
1
vote
2answers
37 views

Continuity set of a difference of two upper semi-continuous real functions over a metric space [on hold]

I wanted to know if we can get some properties of the continuity set of a difference of two upper semi-continuous real functions over a metric space? Or maybe for a restriction?.
0
votes
1answer
17 views

Regarding embeddings of locally convex spaces

If $f:E\rightarrow E'$ is a linear embedding of locally convex topological vector spaces, and $A\subseteq E$ open and convex, can we always find $A'\subseteq E'$ open and convex sucht that ...
0
votes
0answers
35 views

Showing the sphere is not homeomorphic to a torus (my own question!) (or indeed a circle and a washer) - OR puncturing is not continuous

Motivation imagine a rubber sheet extended over the end of a tube, I am saying: "there is no continuous transformation that can retract this sheet over the side" - it is common place to talk about ...
1
vote
2answers
61 views

Baire sets in locally compact Hausdorff spaces

(This is a follow-up to Compact $G_\delta$ subsets of locally compact Hausdorff spaces.) Suppose $X$ is a locally compact Hausdorff space. The Baire sets in $X$, denoted by $\mathcal Ba(X)$, comprise ...
5
votes
2answers
141 views

$f:\mathbb R\to\mathbb R$ continuous function. Which of the following sets can not be image of $(0,1]$ under $f$?

Let $f:\mathbb R\to\mathbb R$ continuous function. Which of the following sets can not be image of $(0,1]$ under $f$? A. $\{0\}$. B. $(0,1)$. C.$[0,1)$. D.$[0,1]$. ...
1
vote
1answer
15 views

complete proof of non-separability of D space

Let $D$ be a set of càdlàg functions on $[0,1]$. Define $f_\alpha(\cdot) \equiv 1(\cdot \ge \alpha)$ for $\alpha \in [0,1]$, which is obviously in $D$. Then, if we denote $|| \cdot ||$ as a uniform ...
4
votes
0answers
60 views

Conjecture in continuum theory: my proof attempt

Conjecture. Suppose $X$ is a normal connected space such that every nondegenerate closed subset of $X$ is disconnected. Then every proper subcontinuum of $\beta X$ has empty interior. proof attempt. ...
2
votes
1answer
46 views

Embed Torus into Klein Bottle

Is there a continuous map of the torus into the Klein bottle? Can one do this so that it is locally a homeomorphism (or a complete embedding)? My idea is to take the square $[-1,2] \times [-1,1]$ and ...
0
votes
0answers
70 views

Why does this imply that two homotopic maps $h,k:S^1→ S^1$ must have the same degree?

I want to show that if two maps $h,k:S^1→ S^1$ are homotopic, then they have the same degree. We define the degree of a continuous map $h: S^1 \rightarrow S^1$ as follows: Let $b_0$ be the point ...
3
votes
1answer
54 views

Theorem 4.22 from baby Rudin: continuity and connectedness

I have some parts that I don't understand from the given proof. The theorem is: If $f$ is a continuous mapping of a metric space $X$ in to a metric space $Y$, and if $E$ is a connected subset of $X$, ...
1
vote
2answers
43 views

Proving an operation of interior is a set of open sets.

I am looking at General Topology notes on Alternative Ways of Defining Topology and come up with this questions: If $\iota : \mathscr P (X) \to \mathscr P (X)$ is an operation of interior, then $\tau ...
0
votes
2answers
38 views

Conditions on $A,B$ that are inherited by $A+B$.

Let $A,B$ be subsets of $\Bbb{R}$. Which of the following is false: If $A,B$ are bounded, then so is $A+B$. If $A,B$ are open, then so is $A+B$. If $A,B$ are closed, then so is $A+B$. ...
2
votes
1answer
44 views

Proof that a continuous function from the unit ball to itself without fixed points implies existence of retract from unit ball to unit sphere

Assume $f:B_{1}\to B_{1}$ (where $B_{1}$ is the closed unit-ball in $\mathbb{R}^{n}$ ) is a continuous function that has no fixed points I need to construct a function $g:B_{1}\to B_{1}$ which ...
0
votes
1answer
19 views

Proving the basis formulation of the topology of compact convergence

Definition. Let $(Y,d)$ be a metric space; let $X$ be a topological space. Given an element $f$ of $Y^X$, a compact subspace $C$ of $X$, and a number $\epsilon \gt 0$, let $B_C(f,\epsilon)$ denote ...
2
votes
0answers
49 views

Compact $G_\delta$ subsets of locally compact Hausdorff spaces

Suppose $X$ is a locally compact Hausdorff space and $F$ is a closed subset thereof. Then of course $F$ is also locally compact and Hausdorff. Let $K$ be a subset of $F$, and suppose that $K$ is a ...
1
vote
2answers
67 views

If $(X,d')$ is totally bounded and $d'$ and $d$ are topologically equivalent then $(X, d)$ is separable

I am trying to write something similar to the proof of If $(X,d)$ totally bounded then $(X,d)$ separable but I dont know how to use topological equivalence here. Any help?
0
votes
3answers
38 views

Is this true: Two sets are separated iff they are disjoint and open?

Two sets A,B are separated if no point of A lies in the closure of B and no point of B lies in the closure of A. I know that if A and B are disjoint and open then they are separated, but is it also ...
2
votes
4answers
76 views

Definitions of Open Set and Topological Space

I having trouble understanding two basic concepts in topology: (1) the definition of an open set and (2) the definition of a topological space. OPEN SETS Consider a disc $A$ in $\mathbb{R}^{2}$. We ...
3
votes
2answers
58 views

A wheel with $n$ is not homeomorphic to a wheel with $m$ spokes

Define $W_{n}=\mathbb{S}^{1}\cup\bigcup_{k=0}^{n-1}\left\{ te^{\frac{2\pi k}{n}i}\mid t\in[0,1]\right\}$ $W_{n}$ is called "a wheel with $n$ spokes". Prove that $W_{n} \ncong W_{m}$ for $n\ne m$. ...
0
votes
0answers
24 views

Showing that an evaluation map is continuous

This is a problem from Munkres' Topology 43.8 If $X$ and $Y$ are spaces, define e : $X \times \mathscr {C}(X,Y) \to Y$ by the equation e($x,f$) $= f(x)$; the map e is called the evaluation map. ...
1
vote
2answers
31 views

A metric space is complete if for some $\epsilon \gt 0$, every $\epsilon$-ball in $X$ has compact closure.

This is a problem from Munkres' Topology. Let $X$ be a metric space. (a) Suppose that for some $\epsilon \gt 0$, every $\epsilon$-ball in $X$ has compact closure. Show that $X$ is complete. (b) ...
2
votes
0answers
21 views

Circloid with non-empty interior

By a "circloid" I mean a continuum in the plane which separates the plane into exactly two components and minimal with respect to these properties, i.e. has no proper subcontinuum which separates the ...
2
votes
1answer
56 views

Boundary of Boundary of a set?

I have reading about the closure interior and boundary operators in a topological space. I have been thinking about the following: If $ b $ denotes boundary operator and $c$ , $i$ and $k$ denote ...
1
vote
2answers
66 views

The function $\frac1x$ is an homeomorphism

I have the function $f:(0,+\infty)\rightarrow (0,+\infty)$ defined by $f(x)=\frac1x$ I want to prove that $f$ is an homeomorphism. So I have that $f$ is surjective or onto by definition and that $f$ ...
0
votes
1answer
45 views

How can I show $G_0$ and $G_1$ are conjugate subgroups?

Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the induced homomorphism from the fundamental group of $E$ to the fundamental group of $B$. Let $e_o$ and $e_1$ be points in E ...
4
votes
2answers
64 views

How do I show that $S^1$ is the suspension of $S^0$?

How do I show that $S^1$ is the suspension of $S^0$? I have all the definitions here, I'm just bad at applying them. The suspension of a topological space $X$ is the quotient $CX / (X × ${$1$}$)$, ...
1
vote
1answer
41 views

Is the suspension space contractible?

Let $X$ be a topological space. The suspension of $X$, denoted $ΣX$, is the quotient $CX / (X × ${$1$}$)$, where $CX$ is the cone on $X$, the quotient space $(X × [0, 1]) / (X × ${$0$}$)$. Is $ΣX$ ...
0
votes
1answer
49 views

How to prove the Cone is contractible?

Let $\sim$ be an equivalence relation on a topological space $X$ with a subset $A ⊂ X$ such that the equivalence classes are: $A$ itself and Singletons $\{x\}$ such that $x ∉ A$. Then define $X/A$ ...
2
votes
1answer
65 views

A direct proof that a compact metric space is sequentially compact

I am looking for a direct proof (not by contradiction) that a compact metric space is sequentially compact, ie constructing a converging subsequence from any sequence. Thanks
2
votes
3answers
82 views

What space is this Homeomorphic to?

Let $X$ be a topological space and let $A ⊂ X$. Let $\sim$ be an equivalence relation on $X$ such that the equivalence classes are: $A$ itself and Singletons $\{x\}$ such that $x ∉ A$. Then ...