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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2answers
50 views

Connectivity of a subset of the topologist's sine curve

I have a question about Example 2 of Section 25 (p.160) of Munkres's Topology. Let $S$ be the following subset of the plane $\mathbb{R}^2$: $$S = \{ \ x \times \sin \tfrac{1}{x} \ \colon \ 0 < x ...
1
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1answer
23 views

$\operatorname{fr}(F)= F$, if $F$ is a set without cluster points.

I was reading the following metric spaces all of whose decompositions are metric and in (a) $\Rightarrow$ (d) I have problems about the "clearly $\operatorname{fr}(F)=F$". One side is easy, since $F$ ...
3
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0answers
83 views

Is the Banach space of continuous functions on a compact space with a coutable base separable?

Let $X$ be a compact Hausdorff space with a countable base. Let $C(X)$ be the Banach space of complex valued continuous functions on $X$. Is $C(X)$ separable, i.e. does it have a countable dense ...
0
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3answers
64 views

At the point $\sqrt{2}$ in the real line, does *every* n-ball around that point contain a rational?

Is it trivial to prove? Obviously a ball of some radius will contain a rational number, but what about for all $\varepsilon > 0$ ?
2
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1answer
60 views

Definitions from topology

I'm reading some papers on the unknotting problem in Knot theory and am running into some notation I don't know (my exposure to topology is minimal, but I have seen it in Analysis courses, Algebra, ...
0
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1answer
31 views

Subset of a normal space

Given X a normal space, and a subset $A \subset X$ not closed. Does it imply A is not normal? I understand it does not, Can someone provide me a counterexample?
2
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0answers
83 views

Can a metric subspace be completely covered by balls after a finite number of steps?

Let $X$ me a metric space with distance $d$ and $A$ be a subspace of $X$. Let $B_\varepsilon(x)$ be the open ball centered in $x$ with radius $\varepsilon$, i.e. $\{y\in X\mid d(x,y) < ...
2
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0answers
32 views

Inverse limit of irreducible spaces

Let $(X_{i})_{i \in \mathbb{N}}$ be an inverse system of topological spaces. Assume that each of the $X_{i}$ is irreducible. Then is it true that $\projlim X_{i}$ is also irreducible? I read in a ...
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1answer
52 views

Making sense of the expression $\lim_{x \rightarrow k^+}f(x)$ using filters, and a reference request.

I don't know much filter convergence, so this is addressed to those who do. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ denote a function. In elementary real analysis, we often write: $$\lim_{x ...
0
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0answers
34 views

Proving that local base determines topology.

Let $(X,\tau)$ be a topological vector space and $\mathcal{B}$ a collection of neighborhoods of $0$ such that every neighborhood of $0$ contains a member of $\mathcal{B}$ (that is, $\mathcal{B}$ is a ...
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0answers
34 views

Prob. 9, Sec. 24 in Munkres' TOPOLOGY, 2nd ed: How to achieve this? [duplicate]

Let $A$ be a countable subset of $\mathbb{R}^2$. How to show that $\mathbb{R}^2 - A$ is path-connected? My effort: Let $A = \{ a_1, a_2, a_3, \ldots \}$, where $a_n = (\alpha_n, \beta_n) \in ...
0
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2answers
154 views

Prove that each component of $X$ is a closed subset of $X$.

Definition: Let $X$ be a topological space and let $\sim_C$ be the equivalence relation on $X$ defined by $x \sim_C y$ if $x$ and $y$ lie in a connected subset of $X$. The components of $X$ are the ...
1
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1answer
36 views

Prob. 4, Sec. 24 in Munkres' TOPOLOGY, 2nd ed: How to verify the supremum property?

Let $X$ be an ordered set in the order topology. Suppose that $X$ is connected. How to show that $X$ is a linear continuum?
1
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1answer
45 views

Union of connected sets

$\forall \beta \in I$, $A_{\beta }$ is connected, and $\left ( \bigcup_{\alpha < \beta }A_{\alpha } \right )\cap A_{\beta }\neq \varnothing$ . Is $\bigcup_{\alpha \in I}A_{\alpha } $connected? For ...
0
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2answers
70 views

Subspace of Lindelöf space is not Lindelöf: Example

The Munkres' topology book provides Example 30.5 (p.193, 2nd Ed) for a subspace of a Lindelöf space that need not be Lindelöf as follows: The ordered square $I_0^2$ is compact; therefore it is ...
1
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1answer
34 views

Let $f:(X, \mathfrak T_X) \rightarrow (Y, \mathfrak T_Y)$ be a continuous function. Then $f(Cl(A) = Cl(f(A))$.

Let $f:(X, \mathfrak T_X) \rightarrow (Y, \mathfrak T_Y)$ be a continuous function. Then $f(Cl(A) = Cl(f(A))$. My definition of closure is: Let $(X,\mathfrak T)$ be a topological space and let $ A ...
8
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1answer
74 views

Informal interpretation of meager sets

I've been wondering if there is a nice informal interpretation of meager sets akin to the respective interpretations I give below to other notions of "small" sets. The general setup to tease out ...
0
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0answers
18 views

A non-zero linear functional can induce a positive linear functional?

A non-zero linear functional can induce a non-trival positive linear functional? $X$ is a topological sepace, if $ f:C_{c}(X)--->R $, is a linear functional, and $f$ is not always equal to zero, ...
113
votes
5answers
3k views

Why can't differentiability be generalized as nicely as continuity?

The question: Can we define differentiable functions between (some class of) sets, "without $\Bbb R$"* so that it Reduces to the traditional definition when desired? Has the same use in at least ...
0
votes
1answer
20 views

Completely Regular Spaces and Embeddings

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. We were going over separation axioms in class when assigned the following problem. Given ...
0
votes
1answer
27 views

Topology induced by a closed-finite topology

Let $(X, \tau)$ be a topological space where $\tau$ is the closed-finite(co-finite) topology. Consider $A \subset X$, is the topology$\tau_{A}$ induced on $A$ by $(X, \tau)$ going to be closed-finite? ...
2
votes
1answer
70 views

Let $M\subseteq \mathbb{R}^k$: Manifold topology vs. trace topology

I'm confused about the topology of submanifolds of $\mathbb{R}^n$: Let $M$ be such a $k$-manifold (say, the circle $S^1$, of dimension $1$, embedded in say $\mathbb{R}^7$); the topology of such a ...
2
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0answers
54 views

Stokes Theorem Manifold with Corners Proof

I'm working through the proof for Stokes' Generalized Theorem for Manifolds and have a questions about corners. I've seen several proofs for manifolds with corners by creating diffeomorphisms to ...
3
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2answers
56 views

Example 5, Sec. 24 in Munkres' TOPOLOGY, 2nd ed: Is this map always continuous?

Let $(X, \Vert \cdot \Vert)$ be a given normed space that has elements other than the zero vector $\theta_X$. And let $T \colon X-\{\theta_X \} \to X$ be defined by $$T(x) \colon= \frac{1}{\Vert x ...
0
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1answer
44 views

Example 2, Sec. 24 in Munkres' TOPOLOGY, 2nd ed: How to show this set to be a linear continuum?

Let $X$ be a well-ordered set, let $[0,1)$ denote the half-open interval (open from the right) on the real line, and let $X \times [0,1)$ have the dictionary order. Then how to show that $X \times ...
1
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2answers
99 views

Examples about compactness

Compactness implies countably compactness which in turn implies limit-point compactness. Sequentially compactness implies limit point compactness. $Z_{+} \times \{0,1\}$ with two-point indiscrete ...
0
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0answers
29 views

About a map between two topological manifolds with different dimensions

Let $M_1$ be a $n$-dimensional topological manifold and let $M_2$ be a $m$-dimensional topological manifold, such that $m>n$. Moreover, let $U\subset M_1$ be an open set and let $f:U\rightarrow ...
2
votes
1answer
27 views

On general topological spaces and $C(X, \mathbb R)$ , where for closed sets $A,B$ in $X$ , $I_A=I_B \implies A=B$

Let $X$ be a metric space and $C(X, \mathbb R)$ be the ring of all real valued continuous functions from $X$ . For $A \subseteq X$ , let us define $I_A :=\{f \in C(X, \mathbb R) : f(x)=0 , \forall x ...
1
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1answer
74 views

What is the stone space $S_n(T)$ for a theory with infinitely many equivalence classes, each class infinite?

Let $L$ be the (first-order) language with one binary relation symbol $E$, and $T$ be the $L$-theory asserting that $E$ is an equivalence relation with infinitely many classes, each of which is ...
1
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1answer
85 views

Brouwer's fixed point continuous function

Can anyone point me out the continuous functions without brouwer fixed point's for the following sets $$A = \{x \in \mathbb{R}^2 | x_1,x_2 \geq 0 \text{ and }x_1^2+x_2^2 = 1 \}$$ $$B = \{x \in ...
1
vote
1answer
37 views

Bijectivity of radial projection

So I'm trying to show that the boundary of a simplex is homeomorphic to a sphere, and I want to do it by radial projection. But it's turning out to be surprisingly difficult. Intuitively, it is clear ...
0
votes
2answers
56 views

How can $0$ be an interior point of $[0,1]$ when $\mathbb R$ is given the discrete topology?

Let $\mathbb R$ be topologized with the discrete topology. Then every subset of $\mathbb R$ is clopen. So, for every $A \subset \mathbb R$, $\operatorname{int}(A)=A$. But if $A=[0,1]$, the ...
0
votes
2answers
55 views

Confusion over the concept of “compactness”

I have to prove some stuff that involves the concept of collection, in particular those relating to compact sets. But then I have got this trouble. For example, consider the set of all rationals. If ...
3
votes
0answers
55 views

Prob. $ 9 $, Sec. $ 23 $ of Munkres’ “Topology”, $ 2^{\text{nd}} $ Ed.: How to show this subspace is connected?

Let $A$ be a proper subset of $X$, and let $B$ be a proper subset of $Y$. If $X$ and $Y$ are connected, how to show that $$\left(X \times Y \right) \setminus \left(A \times B \right)$$ is also ...
2
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1answer
55 views

The set of points of continuity of a real-valued function on a metric space is a $G_\delta$ set

Let $f$ be a real-valued function on a metric space $X$. Show that the set of points at which $f$ is continuous is the intersection of a countable collection of open sets. I know lots of other ...
0
votes
2answers
87 views

The distance between two sets does not change if closure is taken

Given $ (X, d)$ a metric space, $ A, B \subset X$, show that $ d(A, B)=d (\overline {A}, B) $. I'm not being able to show that $ d(A,B) \leq d (\overline {A}, B) $. Can anybody help me? The set ...
2
votes
1answer
52 views

Some question about path connectedness

I think that's intuitively evident but I can't prove that the set $\mathbb{S}^n\setminus\{(0,\cdots, 1),(0,\cdots, -1)\}\; (n>1)$ is path connected. Does anyone have a formal argument to prove it? ...
0
votes
2answers
76 views

Regular and non-regular covering spaces of $ \Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1} $.

I tried to draw the regular and non-regular covering spaces of $ \Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1} $. I think the regular covering space is: Is it true? How do you draw the non-regular ...
1
vote
1answer
62 views

Prove that a function is open

Let $X,Y$ metric spaces and $U \subset X , V\subset Y$ open sets. Let $f:U\rightarrow V$ be a homeomorphism. Prove that $f$ is an open map. I need to show that for every open subset of $U′⊂U$, $f(U′)$ ...
1
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1answer
62 views

Let $A = \{1- \frac 1n : n \in \mathbb Z ^+\}$ is closed under certain topologies on $\mathbb R$.

Let $A = \{1 - \frac 1n : n \in \mathbb Z ^+\}$ is closed under certain topologies on $\mathbb R$. I am supposed to figure out if this set is closed under certain topologies. I know that means I ...
0
votes
2answers
85 views

Suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$.

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$. I know this a true statement so now I need to ...
0
votes
0answers
47 views

Operator norm on the space on linear functions between Euclidean spaces.

*I'm reading a text which has a preliminary section on Linear maps. I have come across a conclusion that I can't seem prove by myself. * Let $Lin(\mathbb{R}^m,\mathbb{R}^n)$ be the space of linear ...
2
votes
3answers
249 views

Confusion about concept of basis in point set topology.

I'm afraid that I have a big misunderstanding about the notion of basis in general topology. For a given topology $\tau$ of set $X$, if there is a collection $S \subset \tau$ of open subsets of $X$ ...
15
votes
2answers
207 views

Does the forgetful functor from $\mathbf{TopGrp}$ to $\mathbf{Top}$ admit a left adjoint?

Let TopGrp be the category of topological groups (not necessarily $T_0$) and Top the category of topological spaces. Does the forgetful functor $U:\mathbf{TopGrp}\to\mathbf{Top}$ admit a left ...
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votes
1answer
56 views

Probs. 2 (d) and 2(e) in Supplementary Exercises, Chap. 2 in Munkres' TOPOLOGY, 2nd ed: How are these maps continuous?

Let $S^1$ denote the set of all complex numbers $z$ such that $\vert z \vert = 1$ (regarded as a subspace of the complex plane), and let the map $f \colon S^1 \times S^1 \to S^1$ be defined by $$f(w ...
8
votes
2answers
268 views

Do two closed subsets of $[0, 1]$ with measure $\frac{1}{2}$ intersect?

Let $A$ and $B$ be two closed subsets of $[0,1]$, each with a length of $1/2$. Is it always true that $A\cap B\neq \emptyset$? My intuition is yes, because: Either they intersect in their interior; ...
1
vote
1answer
31 views

Continuity of product maps

Let $J$ be a given (countably or uncountably infinite) index set. Let $\{\ X_\alpha \ \colon \ \alpha \in J \ \}$ and $\{\ Y_\alpha \ \colon \ \alpha \in J \ \}$ be collections of topological ...
2
votes
0answers
43 views

Show if orbit is discrete, the orbit is closed.

Given $\beta <$ the isometry group $\mathbb{R}^2$. Show that if an orbit $\beta_x$ is discrete, then $\beta_x$ is closed. I am just looking for some feedback and critique of my attempt at a ...
1
vote
1answer
55 views

Is there any simply connected polyhedron with a not simply connected face?

According to Wikipedia, For a convex polyhedron or more generally for any simply connected polyhedron whose faces are also simply connected, χ = 2. Is it really necessary to specify here, that ...
0
votes
1answer
65 views

Comparing different topologies on the Hilbert cube $H = \prod_{n \in \mathbb{N}} [0,\frac 1n]$

This is essentially exercise 8(c) from section 20 (p.128) of Munkres's Topology: Let $X$ be the set of all the sequences $(x_n)$ of real numbers such that the series $\sum_{n=1}^\infty x_n^2$ ...