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
votes
2answers
965 views

an homeomorphism from the plane to the disc

Someone asked me to give an explicit homeomorphism between $\mathbb C$ and the unit disc. I gave him the following answer: we look at $\mathbb C$ as $\mathbb R^2$. The map $x\mapsto tang(\pi x/2)$ is ...
1
vote
1answer
134 views

Is there a countable pseudocharacter Hausdorff space X with the property A?

Let X be a Hausdorff space and define the Property A as follows: If $\mathscr{U}$ is a collection of open sets of $X$ that witnesses Hausdorff property of $X$ (= $\forall x,y \in X$, there exist ...
4
votes
2answers
596 views

first countable $\Leftrightarrow$ compact and Hausdorff?

Can someone give me a short sketch of a proof or a space that serves as a counterexample to the fact that every first countable space is characterized by being compact and Hausdorff (or, stronger than ...
2
votes
1answer
70 views

Function from a sphere

This is a very basic question but i have difficulties understanding the following: If I have a function which has as its domain a 1-Sphere $S^1$ then how do I have to imagine such a function? Will ...
3
votes
1answer
278 views

Homeomorphism between two spaces

I am asked to show that $(X_{1}\times X_{2}\times \cdots\times X_{n-1})\times X_{n}$ is homeomorphic to $X_{1}\times X_{2}\times \cdots \times X_{n}$. My guess is that the Identity map would work but ...
13
votes
2answers
441 views

“Planar” graphs on Möbius strips

Is there an easy way to tell if a graph can be embedded on a Möbius strip (with no edges crossing)? A specific version of this: if a simple graph with an odd number of vertices has all vertices of ...
3
votes
1answer
170 views

complete metric space

Prove or disprove: $(A_i)_{i=1}^\infty$ are closed subsets in a complete metric space. Assume that there is an open ball in the $\bigcup\limits_{i=1}^\infty A_i$ , so exists $k$ s.t $A_k$ contains ...
1
vote
1answer
192 views

Double of a manifold

Let $M$ be a connected $n$-manifold with a non-empty boundary. The double of it is given by $$ D(M) = M\,\,\,\cup_f\,\,\, M $$ where $f:\partial M\to\partial M$ is an identity map. I have to show ...
0
votes
1answer
123 views

Follow-up: that the closures of every two open sets must intersect

I was reading through general-topology posts, but I couldn't understand the reasoning behind the answer of this part of a post. I'll restate it here: This concerns the irrational slope topology. That ...
5
votes
3answers
170 views

How do you prove that $\mathsf{bd}(\mathsf{bd}(\mathsf{bd}(W)))= \mathsf{bd}(\mathsf{bd}(W))$

$\newcommand{\bd}{\operatorname{bd}}$Prove that the $\bd(\bd(\bd(W)))=\bd(\bd(W))$ where $W$ is a subset of the topological space $(X,\mathscr{T})$.
12
votes
1answer
1k views

open maps which are not continuous

What is an example of an open map $(0,1) \to \mathbb{R}$ which is not continuous? Is it even possible for one to exist? What about in higher dimensions? The simplest example I've been able to think of ...
1
vote
2answers
114 views

Closure and inclusions

Let $A$ and $B$ be subsets of $\mathbb{R}^n$. If $A$ is open, and $B$ is arbitrary, does one always have the following inclusion $A \cap \operatorname{cl}(B) \subseteq \operatorname{cl}(A\cap B)$. Is ...
2
votes
2answers
261 views

Uniqueness of the Quotient Topology

Let $q:X\to Y$ be a surjective map, where $(X,\tau_X)$ is a topological space. The quotient topology $\tau_q$ on $Y$ is given as $U\in \tau_q$ iff $q^{-1}(U)\in \tau_X$. Suppose that there is ...
4
votes
2answers
698 views

Determine the closure of the set $K=\{\frac{1}{n}\mid n\in\mathbb N\}$ under each of topologies

The questions are the following: Consider the five topologies on the real line $\mathbb R$: $\mathcal T_1$: the standard topology $\mathcal T_2$: the $K$-topology $\mathcal T_3$: the finite ...
0
votes
1answer
220 views

Int (A), Bd(A), A' and Cl(A) in the following cases

The question is following: Identify $\mathrm{Int}(A)$, $\mathrm{Bd}(A)$, $A'$ and $\mathrm{Cl}(A)$ in the following cases: (a) $X=\{a,b\}$ with the discrete topology, $A=\{a\}$. My answer ...
0
votes
1answer
259 views

K-topology satisfies the Hausdorff axiom?

Let R be the set of all real numbers and let K={1/n, n is a natural number}. Generate a topology on R by taking as basis all open intervals (a,b) and all sets of the form (a,b)-K (the set of all ...
12
votes
5answers
2k views

A compact Hausdorff space that is not metrizable

Is there an example of a compact Hausdorff space that is not metrizable? I was thinking maybe the space of continuous functions $f: X \rightarrow Y$ between topological spaces $X, Y$, might work, but ...
7
votes
2answers
217 views

Is $[0,1]^\omega$ a continuous image of $[0,1]$?

Is $[0,1]^\omega$, i.e. $\prod_{n=0}^\infty [0,1]$ with the product topology, a continuous image of $[0,1]$? What if $[0,1]$ is replaced by $\mathbb{R}$? Edit: It appears that the answer is yes, and ...
6
votes
0answers
348 views

Homotopy extension property vs. good pairs

I'm taking a course that uses the book "algebraic topology" by Allen Hatcher. I this book there are two different ways in which a pair (X,A) of a topological space X and a subspace A can be nice: They ...
1
vote
0answers
211 views

Is there a ccc but not separable space $X$ with a zeroset-diagonal, that isn't submetrizable?

Is there a ccc but not separable space $X$ with a zeroset-diagonal, that isn't submetrizable? separable = $X$ has a countable dense subset. A space $X$ has a zeroset-diagonal when there is a ...
1
vote
1answer
166 views

Continuous injective map

Consider a topological space $X$ and a subset $Z \subset X$. Assume we are given a continuous injective map (in $X$ topology) $f:Z \to Z$ such that $g: Z \setminus C \to Z$(where $C \subset Z$) and ...
3
votes
1answer
344 views

Long proof of equivalence of subspace and metric topology

Let $(X,d)$ be a metric space and $S\subseteq X$. Let $\tau$ be the topology on $X$ induced by $d$ and $\tau_S$ be the subspace topology on $S$: $$ \tau_S = \{S\cap V:V\in \tau\}. $$ Denote $B_r(x)= ...
4
votes
1answer
313 views

Constructing a sequence that converges to an accumulation point of a set

My question is about characterizing accumulation points in terms of convergent sequences in a set $X$. I take the following definition of an accumulation point: A point $x_0$ is an accumulation ...
0
votes
1answer
141 views

In a regular topology, for any $x$ in an open set $V$, the closure of $V$ is a subset in an open set containing $x$

I want to show that if $X$ is regular then for any $x \in X $ and any open set $U$ containing $x$, there is an open set $V$ s.t. $x \in \overline{V} \subseteq U$. From the regularity definition it ...
2
votes
1answer
290 views

A few clarifications on the irrational slope topology

Yesterday, user t.b. linked me a passage in Counterexamples in Topology. In example 75, linked above, I have a questions on properties 2 and 5. The closure of each basis neighborhood ...
6
votes
1answer
757 views

Does the p-norm converge to the max-norm in some norm

Does the $p$-norm on $\mathbf{R}^n$ converge to the max-norm on $\mathbf{R}^n$ as elements in the space of real valued continuous functions on $\mathbf{R}^n$ endowed with some norm? More precisely, ...
1
vote
1answer
156 views

Is this set of equivalence classes a Tychonoff space?

I'm not sure why the space $(Y,\mathcal{V})$ described below is completely regular. I start by taking $(S,\mathcal{U})$ and defining an equivalence relation $s\sim t\iff f(s)=f(t)$ for every ...
1
vote
1answer
328 views

Is a subset of $\mathbf{R}^n$ closed iff it can be covered by a finite number of closed balls?

A subset $U\subset \mathbf{R}^n$ is open if and only if it can be covered by open balls. (An open ball is a subset of the form $\{x\in \mathbf{R}^n: d(x,y) < r\}$, where $y$ is the origin of the ...
6
votes
1answer
953 views

Interior and boundary points of $n$-manifold with boundary

I'm reading Lee, 'Introduction to Topological Manifolds', 2011. After he introduces $n$-manifold with a boundary An $n$-manifold with a boundary is a second countable Hausdorff space in which any ...
3
votes
1answer
137 views

Definition of a Manifold with a boundary

Lee in his book on topological manifolds says that An $n$-manifold with a boundary is a second countable Hausdorff space in which any point has a neighborhood which is homeomorphic either to an ...
0
votes
0answers
33 views

Definition of a Manifold with a boundary [duplicate]

Possible Duplicate: Definition of a Manifold with a boundary Lee in his book on topological manifolds says that An $n$-manifold with a boundary is a second countable Hausdorff space in ...
0
votes
1answer
167 views

Any continuous function is the composite of a continuous function and surjection onto a Tychonoff space?

I'm trying to verify some topological properties to see that for any topological space $(S,\mathcal{U})$, there is a continuous surjection $\pi\colon S\to T$ for $(T,\mathcal{V})$ a Tychonoff space, ...
3
votes
0answers
145 views

Definition of a complex analytic space

A complex analytic space is a topological space (say, Hausdorff and second countable) such that each point has an open neighborhood homeomorphic to some zero set $V(f_1,\ldots,f_k)$ of finitely many ...
9
votes
1answer
102 views

If a subset of the plane has open intersection with every line, is it open?

Suppose that $U \subset \mathbb{R}^2$ is such that $U \cap L$ is open in $L$ for any line $L \subset \mathbb{R}^2$ where $L$ inherits the subspace topology from $\mathbb{R}^2$ (ie. $L \cong ...
2
votes
1answer
111 views

Is this weird looking topology on a subset of the rational plane $T_2$?

I'm investigating a topology on that rational plane $\mathbb{Q}^2$, with a subbase which I've had a hard time getting my hands on. Suppose $X=\{(x,y)\in\mathbb{Q}^2\mid y\geq 0\}$ be the given ...
12
votes
3answers
514 views

When is a sequence $(x_n) \subset [0,1]$ dense in $[0,1]$?

Weyl's criterion states that a sequence $(x_n) \subset [0,1]$ is equidistributed if and only if $$\lim_{n \to \infty} \frac{1}{n}\sum_{j = 0}^{n-1}e^{2\pi i \ell x_j} = 0$$ for ever non-zero integer ...
0
votes
1answer
114 views

a construction of a continuous function from a normal space to [0,1]

Let $A$ be closed in $X$ , where $X$ is a normal space, and $A$ is also a countable intersection of open sets. Prove that there exist a continuous function such that $$ \eqalign{ & f\left( x ...
5
votes
2answers
162 views

differentiability and continuity

I have a question which is quite "candid" and for which I can hardly formalize properly every concepts but nonetheless, I was wondering if there were some topoligical and/or algebraic structure ...
4
votes
1answer
291 views

Topological terminology: neither open nor closed

In topology, a set which is both open and closed is often called "clopen." Is there a commonly used term for sets which are neither open nor closed?
35
votes
12answers
12k views

Real life applications of Topology

The other day I and my friend were having an argument. He was saying that there is no real life application of Topology at all whatsoever. I want to disprove him, so posting the question here What ...
6
votes
1answer
270 views

Why does having fewer open sets make more sets compact?

In functional analysis and in PDE theory, we are interested in proving existence results. Such results are generally obtained on some compact space for some given topology. And this is the reason why ...
4
votes
1answer
429 views

Is a metric space perfectly normal?

I typically like to practice my knowledge on a specific concept by doing proofs using one definition of a term, and then doing the same proofs using an equivalent definition (without inducing the ...
3
votes
2answers
303 views

Countable complete set of limit points

Let $(X,d)$ be a metric space with $X$ - countable and such that for any $x\in X,r>0$ there exists $y\in B(x,r)$, $y\neq x$. Can $X$ be complete? I failed to prove that it cannot as well as to ...
1
vote
1answer
84 views

an explicit example of a function with a local strictly maximum dense set

This problem looks very difficult )= Construct a continuous function, such that it set of strictly local maximum points, is the set of rationals.
17
votes
1answer
2k views

Example of a Borel set that is neither $F_\sigma$ nor $G_\delta$

I'm looking for subset $A$ of $\mathbb R$ such that $A$ is a Borel set but $A$ is neither $F_\sigma$ nor $G_\delta$.
2
votes
0answers
53 views

Is the functor associating a bundle with a structure group to a principal bundle faithful?

Consider a (Cartan) principal G-bundle $\xi: X \to B$, and a left $G$-space $F$. We construct the bundle $\xi[F]: X_F \to B$ associated with $\xi$ with a fiber $F$ as usual. Now for each morphism $(u, ...
4
votes
3answers
371 views

Is total boundedness a topological property?

If a metrizable topological space is totally bounded with one metric, is it totally bounded with all others? A related, stronger question: if every metrization of a topological space is bounded, are ...
3
votes
1answer
176 views

Dense *-subalgebras of C*-algebras and their intersections with sub-C*-algebras

Consider the following question: Let $A$ be a normed space containing a closed subset $B\subseteq A$ and a dense subset $D\subseteq A$. Is $B \cap D$ necessarily a dense subset of $B$? My conclusion ...
4
votes
1answer
162 views

Existence of Homeomorphism?

Is there any homeomorphism between $(X,T^1)$ and $(X,T^2)$ where $T^1$ and $T^2$ are topologies on X such that $T^1$ is a proper subset of $T^2$.
3
votes
2answers
264 views

Is the sheaf of locally constant functions flasque?

Quick question, is the sheaf of locally constant functions flasque?