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; ...

learn more… | top users | synonyms (3)

1
vote
0answers
28 views

Topological spaces from compact Hausdorff zero dimensional spaces

I saw a construction of general topological spaces using compact Hausdorff zero dimensional topological spaces, but I have no clue now of the construction or reference to this. I would be thankful if ...
2
votes
1answer
30 views

variations of Kuratowski closure complement theorem

I have been reading about the Kuratowski closure-complement theorem from the paper "THE KURATOWSKI CLOSURE-COMPLEMENT THEOREM by B.J. Gardner and M. Jackson'. It states that: If $(X,\tau)$ is a ...
-2
votes
1answer
25 views

How can I prove that it isn't a compact space [closed]

Let $X=N$ and $B$ is a base for topology $τ(B)$ on $N$ . $B$={φ,{0,1,2,3},{4,5,6,7},{8,9,10,11},........} how can I prove that ($N$,$τ(B)$) is not compact space
2
votes
1answer
18 views

Closure of intersection with vector subspace

I am confused with the footnote on page 198 of http://www.ma.huji.ac.il/~razk/iWeb/My_Site/Teaching_files/TVS.pdf Essentially: Let $X$ be a topological vector space and $Y$ a finite-dimensional ...
1
vote
2answers
47 views

Question about Rudin's example of topological space

I began reading Rudin's Real and Complex Analysis, and I have a question about the following: Rudin defines a topology $T$ in a set $X$ as the collection of subsets of $X$ such that (i) empty set ...
1
vote
2answers
54 views

Prove or disprove that this function is continuous

If $f(x,y)$ is a real valued continuous function defined in $A \times B$ where $A$, $B$ are compact sets in $\mathbb R^n$ and $\mathbb R^m$ respectively. Let $g(x)=\min_{y \in B}f(x,y)$. Prove or ...
2
votes
1answer
33 views

Why not $(a,b)$ is not possible to define $\rho(f, g)$?

According to C.Adam's Topology: I don't know about compactness, but before introducing compactness in this book, in one of exercises it is asked: "Explain why we cannot generally define $\rho(f, g)$ ...
1
vote
2answers
46 views

Show that $d_V$ is a metric

Problem: For points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ in $\mathbb{R}^2$ define: $d_V(p,q) = \begin{cases}1 & p_1\neq q_1 \ or\ |p_2 - q_2|\geq 1 \\ |p_2 - q_2| & p_1= q_1 \ and\ |p_2 ...
3
votes
1answer
39 views

Show an $R$-module is a direct limit

This is a scenario I've encountered in my class on $p$-adic L functions. Let $G$ be a profinite group which is the inverse limit of a system $(G_i, f_{ij})$ of discrete finite topological groups. ...
4
votes
0answers
37 views

Elementary proof of compact space = exhaustible space?

(This is a repost of a question I asked last year on cs.stackexchange.) The work of Martín Escardó has demonstrated close parallels between classical topology on one hand and computability on the ...
1
vote
1answer
30 views

Suppose a 2-adic metric is defined. Showing that if $d(x,y)$ has a midpoint, then $x=y$

Let $\mathbb{Z}$ be the integers. Recall 2-adic metric $$ d(x,y) = \begin{cases} 0 & x=y \\ \frac{1}{2^{n}} & x \ne y\ \text{and}\ 2^{n} \text{is the largest power of 2 that ...
3
votes
0answers
31 views

Topological features (and / or definition) of homology

I am coming to grips with basic homological algebra as of late, in order to better understand my own subject, that of the study of language. The thing is, I have recently read in some handbook that ...
1
vote
2answers
31 views

Does every ball of boundary point contain both interior and exterir points?

My question is If $x$ is a boundary point of $S$ ($S$ is subset of $R$), does every ball of $x$ contain both interior points and exterior points of $S$? I think this is false. Since $R$ is union of ...
3
votes
2answers
42 views

Fundamental group of $\mathbb{R}^n\backslash \{0\}$

I am wondering about what the fundamental group of $\mathbb{R}^n \backslash \{0\}$ or more generally $\mathbb{R}^n \backslash U$ where $U$ is a subset of $\mathbb{R}^n$ for $n>1$. For $n=1$ I ...
2
votes
0answers
31 views

Question about contractible set .

Please if i have a contractible and closed set $A$ in $X$ thene $A$ is closed and there existe a continuous function $H:[0,1]\times A\rightarrow X$ such that $H(0,u)=u, H(1,u)=p\in X.$ If i ...
3
votes
0answers
35 views

Defining a topological relationship between two objects

I am looking for a mathematical definition/description of the following relationship between two objects. It's similar to a knot (as in topology) but between two objects. I've found a similar problem ...
1
vote
1answer
27 views

If X is limit point compact space,which is T1,then X is countably compact.

Countably compact means : every countable open covering contains a finite subcollection that covers it. Limit point compact means: every infinite set contained in it has a limit point. In T1 space ...
7
votes
2answers
151 views

How to find the inverse arc in the configuration space

The following Figure shows the function from configuration space (Torus) to operational space (Annulus). There is a naturally defined continuous function from configuration space $(\theta_A, ...
1
vote
1answer
23 views

Set of all real numbers with the Scott topology

It is known that a space $X$ is compact iff every net in $X$ has a cluster point. Let $\sum\mathbb{R}$ be set of all real numbers with the Scott topology. I know that $\sum\mathbb{R}$ is not compact. ...
1
vote
1answer
36 views

How to show that this set is closed in $\mathbb{R}^n$?

For an open set $\Omega\subseteq\mathbb{R}^n$, let $K_j$ be the set of points $x$ of $\Omega$ such that $\text{dist}(x,\partial\Omega)\geq1/j$ and $|x|\leq j$. Question : Why is $K_j$ closed ? ...
2
votes
1answer
50 views

$\mathfrak{Top}$ and injective objects

My question is very simple. Let $\mathfrak{Top}$ be the category of all topological spaces and continuous functions between them. Does such category have enough injectives? Is there a simple way to ...
1
vote
2answers
35 views

Induced subgroup of $\pi_1(S^1)$ by $p_n$

Consider the following covering map $p_n: S^1 \to S^1, z \mapsto z^n$. Why is the subgroup of $\pi_1(S^1)$ induced by $p_n$ isomorphic to $n\mathbb{Z}$? I know that $\pi_1(S^1) \cong \mathbb{Z}$ but ...
0
votes
1answer
31 views

There is no metric d,so that (Q,d) is a connected space [duplicate]

Can anyone prove this? There is no metric d,so that (Q,d) is a connected space Q are rationals.
4
votes
1answer
66 views

Locally Compact Spaces: Characterizations

For Hausdorff spaces the following are equivalent: Every point admits a compact local base. Every point admits a compact neighborhood. Every point admits a precompact neighborhood. Every point ...
3
votes
3answers
52 views

Embeddings are precisely proper injective immersions.

We call a map $f: X \to Y$ between topological spaces proper if $f^{-1}(K)$ is compact for all compact $K \subset Y$. Where can I find a reference that embeddings are precisely proper injective ...
3
votes
2answers
95 views

Construct a set of real numbers whose limit points comprise the set of integers $\mathbb{Z}$

My thought process is the following: Let $S=\{ m + \frac{1}{n}| m \in \mathbb{Z},n \in N \}$. Then I need to show that the limit points of $S$ are indeed the integers and that these are the only ...
0
votes
0answers
43 views

Cantor set--nowhere dense, complete

I can't figure out this out. Cantor set is closed in $\mathbb{R}$. $\mathbb{R}$ is a complete metric space. Every closed subset of a complete space is also complete; thus, so is the Cantor set. ...
1
vote
0answers
30 views

Separable spaces and functions that separate points

In a metric space, does existence of a function that separates points imply that the space is separable and conversely? I'm just a baby Rudin student. Thanks in advance for every hint.
0
votes
1answer
40 views

Can interior set or exterior set be empty?

I am trying to prove or disprove that if x is a boundary of S in R, then every ball B(x) contains both interior point of S and exterior point of S. I am trying to think of counter example, and one ...
1
vote
1answer
72 views

If $E$ is a closed set there exist a set $S$ such as $E=S'$

In "Elementary Real Analysis" by Thomson-Bruckner p.190 I did the following exercise: (we're working on $\mathbb{R}$ and elementary topology on that set) One of Cantor's early results in set ...
1
vote
1answer
54 views

Example of $I$-adic topology of submodule not matching subspace topology?

I'm reading about the $I$-adic topology on $M$ for $R$ a commutative ring, $I$ an ideal of $R$ and $M$ an $R$-module. The references I'm reading don't provide examples, but they say that if $N$ is a ...
1
vote
1answer
36 views

How does one see connectedness of a covering space?

Something can be proven about the loops (or their possible lifts?) in the base space which will ensure connectivity of the cover?
2
votes
2answers
47 views

Given $S \subset \Bbb{R}$, show $\textbf{int}(S)+\textbf{ext}(S)+\partial S =\Bbb{R}$

The way I proved it is that we knwo R is open so intR=R. For any point in IntS is inside of IntR and any point in ExtS is inside of IntR. any point that is neither intS nor extS is still inside of ...
2
votes
0answers
40 views

In what way is the representation not continuous

http://www.math.u-psud.fr/~fontaine/galoisrep.pdf pp.7-8 Following the definition of a linear representation it states 'if V (an E-vector space) is equipped with a topological structure and if ...
1
vote
1answer
24 views

augmented chain complex

From Hatcher's Algebraic Topology, I know that a continuous map induces a morphism of chain complexes $f :C(X) → C(Y)$ by invariance of homotopy, but how would I show that $f$ also induces a ...
1
vote
2answers
31 views

The set of all exterior points is an open set

Let $S$ be a subset of $R$. Then the set of all exterior points of $S$ is an open set. My proof is as follows: For any element $x$ in $\operatorname{ext}(S)$ (the set of all exterior points ...
2
votes
2answers
121 views

Topology on $Z_p$

let $Z_p$ denote the $p$-adic integers, then it has a topology as a subspace of $\prod_nZ/p^nZ$, where $Z/p^nZ$ is given the discrete topology. (reference I posted before: Why $Z_p$ is closed.) Now ...
0
votes
2answers
47 views

Is my proof that empty set is open and R is open correct?

Claim: The empty set is open. Proof. Assume that the empty set is closed. Then, there must be one point such that any point in its ball is not inside of the empty set. However, the empty set has no ...
0
votes
0answers
27 views

Acyclic model type result [closed]

If $\sigma: \Delta_n \to X$, define $\overline{\sigma}: \Delta_n \to X$ by$$\overline{\sigma}(t_0, \dots, t_n) := \sigma(t_n, \dots, t_0).$$Define a map $T: C_n(X) \to C_n(X)$ by $T(\sigma) := ...
2
votes
1answer
45 views

$\mathbb{A}^n$ with the Zariski Topology is Quasi-Compact.

I want to show that $\mathbb{A}^n$ is quasi-compact. I'm kind of stuck, I really don't know where to go with my proof, so I'll show what I have Proof: So suppose that $\cup U_i$ was an open cover for ...
2
votes
3answers
41 views

Show that the Euclidean Metric is less than or equal to the Taxi-cab metric for $\mathbb{R}^{n}$

I am trying to prove that the Euclidean metric; $(\mathbb{R}^{n},d^{2})$; defined: $$d^2(x,y) = \sqrt{\sum_{i=1}^n(x_i-y _i)^2}. $$ is less than or equal to the Taxi Cab metric; ...
-1
votes
3answers
72 views

Exercise on Metric space

I hve this exercise it is very simple but i don't know how to write the answer Let $A$ be a nonempty set in $(E,d)$, for $\varepsilon>0$ we note $$V_{\varepsilon}(A)=\{x\in E, ...
0
votes
0answers
42 views

How to convert an object into a sphere?

I'm not sure if I understand it enough, but doesn't the Poincare conjecture prove any shape can be turned into a sphere? How would I go about transforming such an object? Like let's say I have a ...
0
votes
1answer
35 views

Openness of inverse limit of quotient mappings

Let $X$ be a topological space. Let $(R_\alpha)_{\alpha \in A}$ be a family of equivalence relations on X, whose index set $A$ is directed. We have canonical mappings $\phi_\alpha : X \to X/R_\alpha$. ...
0
votes
1answer
41 views

Requirements for Mayer-Vietoris

This question might be a duplicate -- but as I don't find an entry (maybe because of the lack of a good keyword) I open this question. Besides, this questions arises when trying to prove Proposition ...
0
votes
1answer
39 views

Check $S\cap T$ where $S =\left\{ x \in \mathbb{R} : x^6 -x^5 \le 100\right\}$ and $T =\left\{x^2-2x : x \in (0,\infty)\right\}$

Let $S=\left\{x\in\mathbb{R} : x^6 -x^5 \le 100\right\}$ and $T=\left\{x^2-2x : x \in (0,\infty)\right\}$. Then check whether or not $S\cap T$ is Closed and bounded in $\mathbb{R}$ Closed but not ...
0
votes
1answer
14 views

Topology generated by a Family of Seminorms as a Initial Topology?

Let $X$ be a set and $\{(Y_i, \mathscr{T}_i)\}_{i\in I}$ be a family of topological spaces and $\{f_i\}_{i\in I}$ a family of mappings $$f_i:X\longrightarrow Y_i.$$ The initial topology on $X$ is the ...
0
votes
1answer
38 views

Separability of the Set of Bounded Functions over [0,1]

I'm working through Neal Carothers' Real Analysis and I'm stuck on trying to show that the set $B$ of bounded, real-valued functions over $[0,1]$ is not separable. The metric of this set is ...
2
votes
1answer
21 views

Question About Assumption Relating to Bolzano-Weierstrass

We had the first day of our topology course today, and the instructor presented the Bolzano-Weierstrass Theorem on $\mathbb{R}$. I took it as a challenge to see if I could do it while he was talking, ...
2
votes
2answers
58 views

Show that $\mathbb{A}^n$ on the Zariski Topology is not Hausdorff, but it is $T_1$

There was an exercise I could not do. So the property is $T_1$ if for every pair of distinct points, $P, Q \in X$, there is an open subset $U$ containing $P$ but not $Q$ and another open subset $V$ ...