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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7
votes
3answers
87 views

What does it mean for a topology on a mapping space to correspond to a type of convergence?

I've been asked to prove that three different topologies on $Y^X = \{f : X \to Y | f \text{ is continuous} \}$ correspond to three different types of convergence, but I don't understand exactly what ...
7
votes
3answers
91 views

Verify that $x\mapsto (\cos(x),\sin(x))$ from the real line to the unit circle is an open map.

(Hi, this is my first question on MSE, please let me know if I'm doing anything wrong. Thanks!) The setup: Let $p:\mathbb{R}\to S^1$ be defined by $x\mapsto (\cos(x),\sin(x))$. Prove that $p$ is ...
1
vote
1answer
77 views

Looking for a homeomorphism $\mathbb{C}P^1 \cong S^2$

I want to show $\mathbb{C}P^1 \cong S^2$ by explicit construction. Everything I tried so far did not work out unfortunately :( Any hints?
0
votes
1answer
42 views

Convergence of a sequence in $l_2$

I am wanting to disprove (show that it is not the case) that for a sequence ${x_n} = x^n$, ($n\in\mathbb N$), that if $(x_i)^n \to x_i$ in $\mathbb R$,then $x^n\to x$ in $\ell_2$. I have gotten as far ...
1
vote
3answers
71 views

Is $S^1$ homeomorphic to $\mathbb{R}P^1$?

I am supposed to construct a homeomorphism of $S^1$ and $\mathbb{R}P^1$ but I am not toally sure that this is even possible. I think I have learned at some point that $$\mathbb{R}P^1=S^1/\{x=-x\}$$ ...
2
votes
1answer
43 views

Compact sets closed in Hausdorff spaces without choice?

An elementary proof that compact sets are closed in Hausdorff spaces involves making arbitrary choices based on the Hausdorff property. Is there a way to avoid invoking choice?
0
votes
1answer
20 views

Show the Boundary of a translation is the translation of the Boundary

Define the translation of a set $E$ as: $$a+E=\{z\in \Bbb R^2: z=a+x,\text{ for }x\in E \} $$ I need to show that $\delta (a+E)= a+ \delta E$, where $\delta$ denotes the boundary of the set. i.e ...
0
votes
2answers
34 views

Understanding the definition of continuity from real analysis

I've stared at and worked with the definition of continuity of a real valued function at a point for many (like $3$) years, but there are some things that have always bothered me about it. First, ...
7
votes
2answers
70 views

Is every path-connected open subset of $\mathbb{R}^2$ homeomorphic to $\mathbb{R}^2$?

It is a standard result that the open ball $$B^2=\{(x,y)\in\mathbb{R}^2:x^2+y^2<1\}$$ is homeomorphic to $\mathbb{R}^2$ itself. Also, distorting $B^2$ by any continuous bijective transformation ...
1
vote
3answers
62 views

Another topology problem help

Let $X$ be infinite with the finite complement topology (a set is open if it is $\varnothing$ or its complement is finite). If $A$ is any infinite subset of $X$, show that the closure of $A$ is $X$. ...
1
vote
2answers
67 views

What are the suggested prerequisites for topology?

I am interested in topology but I don't know if I can learn it without learning something else first. I've done: Algebra 1 and 2 Euclidean Geometry Calculus Is that enough if not please tell me what ...
0
votes
1answer
40 views

The topology on $\mathbb Z_+$ given by $\emptyset, \mathbb Z_+$, and all sets of the form $\{n, n+1, n+2, …\}$

Let $\tau$ be the topology on $\mathbb Z_+$ given by $\emptyset, \mathbb Z_+$, and all sets of the form $\{n, n+1, n+2, ...\}$ for all $n\in \mathbb Z_+$. Give the limit points of $A = \{4, 13, 28, ...
0
votes
1answer
49 views

Open map and adherence

How to prove that $$f :E\rightarrow F ~\text{is open} \Longleftrightarrow f^{-1}(\overline{A})\subset \overline{f^{-1}(A)}, \forall A\subset F$$ where $(E,\tau), (F,\theta)$ are topological spaces. ...
14
votes
2answers
66 views

A typical example of Homeomorphism

The set $\mathbb{R}^2-\{(0,0)\}$ with the usual topology is: (A) Homeomorphic to the open unit disc in $\mathbb{R}^2$ (B) the cylinder $\{(x,y,z)\in \mathbb{R}^3/ x^2+y^2=1 \}$ (C) the ...
0
votes
1answer
17 views

Show that the sequence $(A_n)_{n≥1}$ in $L(l_1)$ does not converge to zero

For any $n ≥ 1$, define a linear operator $A_n : l_1 → l_1$ by $$A_nx = (0, . . . , 0, x_{n+1}, x_{n+2}, . . .), ∀x = (x_1, x_2, . . .) ∈ l_1.$$ Show that For any $x ∈ l_1$, we have $\lim_{n→∞} A_nx ...
2
votes
0answers
45 views

Weight of topology related to sizes of open sets?

I'm wondering if there is a notion relating the weight of a topological space (=minimal base cardinality) to the sizes of its open sets. In particular I'm looking for properties of spaces, whose ...
2
votes
1answer
19 views

Open or closed status of addition of two subsets of a metric space

Question Let A and B be subsets of $R^n$. Define A + B = {a + b | a ∈ A, b ∈ B}. Consider the following sets W = {(x, y) ∈ $R^2$| x > 0, y > 0}, X = {(x, y) ∈ $R^2$ | x ∈ R, y = 0}, Y = {(x, y) ...
0
votes
1answer
36 views

Topological supremum of family of linear topologies

In Hans Jarchow - Locally Convex Spaces 2.4.4 (c) it says: The topological supremum of any family of linear topologies on a fixed vector space is linear. I couldn't find a proof in the book and ...
3
votes
1answer
17 views

Convergence of functions with different domain

Question: Is there a concept of convergence for functions $f_n: D_n \rightarrow X$ with different domains to a function $f: D \rightarrow X$? I know concepts like uniform convergence or almost ...
-1
votes
3answers
57 views

Taking undergrad geometry and topology at the same time

I am enrolled in an undergraduate program in mathematics. What would be the general consensus on taking topology and geometry in the same semester? How similar are the courses? What about taking them ...
1
vote
2answers
24 views

Compact zero-dimensional $T_2$-topologies on $\mathbb{N}$

Let $\tau$ be a compact topology on $\mathbb{N}$ such that for every two points $m\neq n\in \mathbb{N}$ there is a clopen set $U$ containing $m$ but not $n$. Is $(\mathbb{N},\tau)$ isomorphic to ...
2
votes
0answers
45 views

Why $(\cos t, \sin t)$ is not a homeomorphism

Consider the map $f: [0,2 \pi ) \to S^1 \subseteq \mathbb R^2$ defined by $t \mapsto (\cos t, \sin t)$. It is clear that $f$ is continuous and bijective. It is not open and therefore not a ...
1
vote
1answer
19 views

Cardinality of a basis for $\prod _{\gamma<\alpha} [0,1]_\gamma$

Claim for any infinite ordinal $\alpha$ there exists a basis for $\prod _{\gamma<\alpha} [0,1]_\gamma$ of size $|\alpha|$. proof. Each factor $[0,1]$ has a basis of cardinality $\omega$ from ...
1
vote
1answer
81 views

Separability, total boundness and topological equivalence of metrics

The problem is: If $(X,d)$ is a separable metric space then there exists a metric $d'$ that is topologically equivalent to $d$ and such that $(X,d')$ is totally bounded. I know that if ...
1
vote
2answers
33 views

Closed AND open subspaces of a normed vector space

Let $E$ be a finite dimension normed vector space. How can I show that $E$'s only both closed and open (norm-wise) subsets are $\emptyset$ and $E$ ?
0
votes
1answer
34 views

A Question about cardinal numbers and homeomorphism

I'm studying my set theory lessons and I don't know if the following statement about homeomorphism between ordinal spaces is true or not and why. "If $\kappa$ is an infinite cardinal number then ...
-1
votes
1answer
55 views

Lindelöf subspaces in the ordinal space [closed]

How can we describe all Lindelöf subspaces of $$X = \omega_1 \times ( \omega_1+1).$$ Where ω1 is the first uncountable ordinal.
2
votes
0answers
42 views

Is $f$ continuous if for every $p$, there is a sequence $p_n \to p$ such that $f(p_n) \to f(p)$?

Let $(X, d)$ be a metric space and $f : X \rightarrow X$ a function that satisfies the following property: For every $p \in X$ there exists a sequence $\{p_n\}\subset X$ such that $p_n \rightarrow p ...
0
votes
0answers
25 views

Are the face posets of CW-complexes Eulerian?

Suppose we had a CW-complex $X$ with decomposition $X_{i}$ Is its face poset, consisting of cells and covers generated by the attachment of cells to one another, an Eulerian poset? What would be the ...
2
votes
0answers
34 views

Knots which are composed of several strands

In a math textbook and this article in NRICH, some problems deal with a special kind of knots: those which are formed from several strands: The problems ask if a given knot can be formed from just ...
1
vote
1answer
62 views

Manifold which is union of two balls is topologically a sphere

In Petersen's book while proving sphere theorem the following fact has been stated without prove : Let $M$ be a connected $n$ dimensional smooth manifold such that $M=B_{1}\cup B_{2}$ where $B_{i}$'s ...
-1
votes
1answer
22 views

Continiuous functions to the sphere

Let $X=AUB$ be a topological space and $A, B$ be a two closed set of X. Let $f:A\to S^n$ and $g:B\to S^m$ be two continuous functions. Define $h:X\to S^{n+m+1}$ by $$h(x)=(f(x),0,\cdots , 0) ...
1
vote
2answers
39 views

option: $X$ can't be homeomorphic to an open subset of $\mathbb R$ to be false

Let $X$ be an infinite set(topological space) homeomorphic to $X\times X$.The book gives the option option: $X$ can't be homeomorphic to an open subset of $\mathbb R$ to be false I cant figure ...
0
votes
0answers
24 views

Sigma algebra which is not a topology [duplicate]

Can you please provide me with some $\sigma$-algebra which is not a topology?
1
vote
1answer
29 views

Covering space action on an orientable manifold $M$ implies $M/G$ orientable (Hatcher)

I'm trying to solve the following problem from Hatcher (3.3.4) Given a covering space action of a group $G$ on an orientable manifold $M$ by orientation preserving homeomorphisms, show that $M/G$ is ...
1
vote
1answer
32 views

Discrete subspace of $\Bbb{R}$ is countable

Show that any discrete subspace of $\Bbb{R}$ with usual topology is countable. Let $U$ be a discrete subspace of $\Bbb{R}$, for each $x\in U$ choose an interval with rational endpoints ...
1
vote
1answer
23 views

Cofinite Topology: Borel Algebra vs. Power Set

Being curious... Are there uncountable spaces such that any uncountable subset has countable complement: ...
3
votes
0answers
60 views

Topology of the space of “loops” [closed]

I have a question that I'm not even sure I can put into words, but please bear with me! I want to define some sort of "loop space" and I want to understand it's topology enough that I can compare it ...
0
votes
1answer
34 views

A sequence converges if and only if every subsequence converges?

I want to prove this and intuitively it makes sense. But I'm having a hard time coming up with a proof. So if a sequence converges, then we have a natural number for which the distance between all ...
3
votes
2answers
43 views

Open vs Closed Sets for studying topology

Topologies can be defined either in terms of the closed sets or the open sets. Yet most proofs, examples, problems, etc. in standard texts concern the open sets. I would think closed sets are ...
3
votes
1answer
50 views

One point compactification of $[0,1]\times [0,1)$

What is one point compactification of $[0,1]\times [0,1)$? If we draw the figure we see that top line is missing and we've to add just one point to make it compact. So I think triangle will be ...
0
votes
2answers
28 views

Regarding distance between sets in metric topology

Let $(X,d)$ be a metric space For subsets $A$ and $B$ of $X$, define $$d(A, B) = \inf\{d(a, b) : a \in A, b \in B\}.$$ Weather the following statement is true or false? If the intersection of ...
-1
votes
1answer
61 views

Homeomorphism and topology [closed]

I have to prove that $$f~\text{is a homeomorphism}\Rightarrow f~\text{is bijective and}~ \overline{f(A)}\subset f(\overline{A}), \forall A\subset E$$ Where $f:E\rightarrow F$ Help me please Thank ...
1
vote
1answer
40 views

Is $C[0,1]$ locally Compact?

I'm asked to use the function $f_n(x)=nx$ for $0\le x\le \frac{1}{n}$ and $f_n(x)=1$ for $\frac{1}{n}\le x\le 1$. I'm not familiar with Functional Analysis.
5
votes
2answers
196 views

Is second-countability invariant under homotopy equivalence?

I am wondering if second-countability is invariant under homotopy equivalence. If I had to guess I would say so. Intuitively, if we have a countable basis of a space, and then stretch, contract, bend ...
3
votes
2answers
68 views

Show $H_2(M, \mathbb{Z}) = \mathbb{Z^r}$ if $M$ is orientable, $\mathbb{Z^{r-1}} \oplus \mathbb{Z_2}$ if nonorientable

I'm trying to solve this problem from Hatcher 3.3.24. Let $M$ be a closed connected 3-manifold, and write $H_1(M, \mathbb{Z})$ as $\mathbb{Z^r} \oplus T$ where $T$ is torsion. Show that $H_2(M, ...
0
votes
1answer
49 views

compactness in metric spaces

Let $f: X \to Y$ be a continuous, closed function where $X$ and $Y$ are metric spaces. Show that if for every $y \in Y$, the set $f^{-1}(y)$ is compact, then for each compact set $F \in Y$, the set ...
0
votes
1answer
16 views

On the wording of a question related to open cover of sets

I am working on the exercise where the hypothesis is : Let $X$ be a scheme such that there exists affine open subsets $U_i \ (1 \leq i \leq n)$ such that $X = \cup U_i$. Further any two of the $U_i$'s ...
0
votes
0answers
34 views

Homeomorphism between the group of $S(O)_{2}$ and the $S_1$.

During an exam I had to prove the following: "Let there be a dynamical system of $n=2$ dimensions and let the eigenvalues that correspond to it, to be imaginary with their real part equal to zero. ...
3
votes
0answers
40 views

Retracts of $\mathbb{Q}$

Let $A \subseteq \mathbb{Q}$ be a non-empty set. Is $A$ a retract of $\mathbb{Q}$? In other words, is there a continuous map $r: \mathbb{Q}\to A$ such that $r|_A = \mathsf{id}_A$?