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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2
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0answers
25 views

Sufficient parametrizations to cover $S^1 \times S^1$. [on hold]

Explicitly exhibit enough parametrizations to cover $S^1 \times S^1 \subset \mathbb{R}^4$.
0
votes
1answer
25 views

Can two cuboids with different side lengths have the same volume and perimeter?

We know that two rectangles with different side lengths cannot have the same area provided their perimeter is the same. But can two cuboids with different side lengths have the same volume and ...
0
votes
0answers
16 views

Let $\mathfrak T_X = \{f^{-1} (U) : U \in \mathfrak T_Y\}$ then $\mathfrak T_X$ is a topology on X. False?

Let $f :X \rightarrow Y$ be a function and suppose that $\mathfrak T_Y$ is a topology on $Y$. Let $\mathfrak T_X = \{f^{-1} (U) : U \in \mathfrak T_Y\}$ then $\mathfrak T_X$ is a topology on X. ...
0
votes
0answers
30 views

Function not equal a.e. to continous function on real line and on circle

I am looking for a proof of the following fact: Suppose that $H: \mathbb{R} \rightarrow \mathbb{R}$ is a periodic function with period $1$. Suppose further that there is no continuous function ...
0
votes
1answer
14 views

Prehilbert space theorem

Let $X$ prehilbert space and $Y$ be a Banach subspace of $X$ and let $x$ belong to $X$ and the vector $p(x)$ is the orthogonal projection of $x$ on $Y$ . Prove that: $x-p(x)$ belong to the ...
0
votes
1answer
23 views

Why is T continuous?

Let $T\colon X\to X$, with $X=\left\{0,1,2\right\}^{\mathbb{Z}}$, desribe the following dynamics: 1 becomes 2 2 becomes 0 0 becomes 1 if at least one of its two neighbours is 1, otherwise it remains ...
-1
votes
0answers
10 views

Hilbert space theorem

Let X prehilbert space and Y be a banach subspace of X and let x belong to X and the vector p(x) is the orthogonal projection of x on Y . Prove that: x-p(x) belong to the orthogonal complement of Y ...
-1
votes
1answer
21 views

Convergence in the Tychonoff topology on $\mathbb{R}^\mathbb{R}$.

This is Example 10.2(b) (p.70) in Willard's General Topology: In the product space $\mathbb{R}^{\mathbb{R}}$, a sequence $f_n$ converges to $f$ iff $f_n(x) \rightarrow f(x)$ for each $x \in ...
0
votes
1answer
16 views

If f is a continuous map from X to Y with X limit point compact, does it follow that f(X) is limit point compact?

I'm looking for proof verification/help for the title question. Here is what I have now: Let $f:X \rightarrow Y$ be continous with $X$ limit point compact. Let $V$ be an infinite subset of $f(X)$. ...
1
vote
1answer
39 views

Example 2, Sec. 25 in Munkres TOPOLOGY 2nd ed: Is this subspace also connected?

The topologist's sine curve is the closure $\overline{S}$ of the subset $S$ of $\mathbb{R}^2$, where $$S\ = \ \{ \ x \times \sin \frac{1}{x} \ \colon \ 0 < x \leq 1 \ \}.$$ So $$\overline{S} = S ...
-2
votes
0answers
11 views

let X be a metric space :X->R be a continous function. G be a graph of f.Then G is homeomorphic to 1. X 2. R 3. X x R 4. R x X [on hold]

let X be a metric space :X->R be a continous function. G be a graph of f.Then G is homeomorphic to 1. X 2. R 3. X x R 4. R x X
0
votes
0answers
13 views

Example 3, Sec. 25 in Munkres' TOPOLOGY, 2nd ed: Why is the topologist's sine curve not locally connected?

The topologist's sine curve is by definition the closure $\overline{S}$ in $\mathbb{R}^2$ of the set $S$ given by $$S = \{ \ x \times \sin \frac{1}{x} \ \colon \ 0 < x \leq 1 \ \}.$$ To show that ...
3
votes
2answers
38 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
vote
1answer
21 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
votes
0answers
17 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
votes
3answers
47 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 $\varepsilon > 0$ ?
1
vote
1answer
46 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
votes
1answer
18 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
votes
0answers
60 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) < ...
1
vote
0answers
23 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 ...
0
votes
2answers
19 views

Bijective continous from R to closed half interval

Can we have bijective continous function from Set of real number R to closed half interval 0 to infinity.
1
vote
1answer
38 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
votes
0answers
21 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 ...
0
votes
0answers
27 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 ...
2
votes
0answers
36 views

The set of non-wandering points of a transformation on $\{0,1,2\}^{\mathbb{Z}}$

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ with $T\colon X\to X$ describing the following dynamics: $1$ becomes $2$, $2$ becomes $0$, and $0$ becomes $1$ if at least one of its two neighbours is a ...
0
votes
2answers
23 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
vote
1answer
27 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?
0
votes
1answer
26 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 ...
-7
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0answers
47 views

GENERAL TOPOLOGY [on hold]

1.Can you give me the counter-example for the theorem "Every compact space is countably compact" . 2.Can you give me the counter-example for the theorem "Every compact subset of a Hausdorff space is ...
0
votes
2answers
15 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
vote
1answer
23 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 ...
3
votes
0answers
23 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 ...
89
votes
4answers
2k views

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

I was a little bit dissapointed when I learned to differentiate on manifolds. Here's how it went. A younger me was studying metric spaces as a first unit in a topology course, when a shiny new ...
0
votes
1answer
16 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
17 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
27 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
votes
0answers
26 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
votes
2answers
52 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
votes
1answer
24 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
vote
2answers
43 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
votes
0answers
20 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
22 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 ...
0
votes
1answer
41 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
vote
1answer
63 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
15 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
48 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
43 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
26 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
votes
1answer
34 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
51 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 ...