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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3answers
276 views

How to prove the group of automorphisms of $S^1$ as a topological group is $\mathbb Z_2$?

The title basically says it all. How does one prove the group of automorphisms of $S^1$ (the unit circle in $\mathbb C$), as a topological group, is $\mathbb Z_2$? I was surprised not to find the ...
3
votes
1answer
555 views

Is the product of Polish spaces Polish?

If $T_t=(E_t,\tau_t)$ are Polish, $t\in I$, is $T^I:=(\prod_{t\in I}E_t,\prod_{t\in I}\tau_t)$ Polish? ($\prod_{t\in I}\tau_t$ being the product topology.) If this is not the case, what extra ...
3
votes
2answers
80 views

Continuous mapping from open set to open set

Suppose we have two open and bounded sets, $\Omega_1,\Omega_2 \in \mathbb{R}^2$. Is there a continuous function $\textbf{f}$ mapping $\Omega_1$ onto $\Omega_2$? \begin{align*} \Omega_1 & = \{(x,...
3
votes
1answer
622 views

Finite unions and intersections $F_\sigma$ and $G_\delta$ sets

Why is the intersection of finitely many $F_\sigma$ sets an $F_\sigma$ set, and the union of finitely many $G_\delta$ sets a $G_\delta$ set?
3
votes
1answer
715 views

Continuous image of the intersection of decreasing sets in a compact space

Suppose $B_{\epsilon}$ are closed subsets of a compact space and $B_{\epsilon} \supset B_{\epsilon'} \quad \forall \epsilon > \epsilon'$. Furthermore, $B_0 = \bigcap_{\epsilon>0} B_{\epsilon}$. ...
3
votes
2answers
83 views

Lie group step in proof

Let $X_e,Y_e \in T_eG$ be vectors and $G = GL(n).$ Then the right translation is given by $Y_g = Y_e g$ and $X_g = X_e g.$ Now, I have a proof showing that $[X_e,Y_e] \in T_eG$ is the element ...
3
votes
1answer
1k views

Proof that Sorgenfrey plane is not normal using points x × (-x)

I'm making Exercise 9 of paragraph 31 in Munkres, which is a proof that the Sorgenfrey Plane $\mathbb{R}_l^2$ is not normal. I'm having trouble on part c of the question. The full question is: Let $A$...
2
votes
2answers
93 views

Why $Z_p$ is closed.

Let $A_n=\mathbb{Z}/p^n\mathbb{Z}$ be a ring and $p$ is prime, $\phi_n: A_n\rightarrow A_{n-1}$ be a natural homomorphism (Elements of $A_{n}$ define in an obvious way elements of $A_{n-1}$). Define $...
2
votes
2answers
1k views

What does order topology over Ordinal numbers look like, and how does it work?

Space of ordinal numbers are one of the favorite examples of my professor in general topology. I quite understand the idea at the base of ordinal numbers (few things or nothing about the concept of ...
2
votes
1answer
1k views

boundary of the boundary of a set is empty

I am learning some stuff about the interior, closure and boundary of sets $A\subset\mathbb R^n$ and I am wondering about the following: 1) $\partial\partial A=\partial A$ ? 2) $\partial\partial\...
2
votes
1answer
229 views

Is there a topological space which is star compact but not star countable?

A topological space $X$ is said to be star compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a compact subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$. A ...
1
vote
2answers
65 views

Examples of a quotient map not closed and quotient space not Hausdorff

Is there any example of a closed relation $\sim$ on a Hausdorff space $X$ such that $X/\sim$ is not Hausdorff? Also, is there any example of a closed relation ~ on a Hausdorff space $X$ such that a ...
0
votes
1answer
94 views

If $p(z,w)=a_0(z)+a_1(z)w+\dots +a_k(z)w^k$ are non constant polynomial.

$p(z,w)=a_0(z)+a_1(z)w+\dots +a_k(z)w^k$ where $a_i(z)$ are non constant polynomials in complex variables with $k\ge 1$. I need know if $$\{(z,w):p(z,w)=0\}$$ which of these are true or false: ...
16
votes
2answers
404 views

A sort of inverse question in topology

Given topological spaces $X$ and $Y$, we often consider the collection of continuous functions, $f: X \rightarrow Y$. My question is, given two sets $X$ and $Y$, and a sub-collection $\{g_{i}\}$ of ...
14
votes
1answer
133 views

Exists homeomorphism which carries each fiber isomorphically to itself, composition?

Let $\mu$ and $\mu'$ be two different Euclidean metrics on the same vector bundle $\xi$. How do I see that there exists a homeomorphism $f: E(\xi) \to E(\xi)$ which carries each fiber isomorphically ...
14
votes
5answers
424 views

Compactness in $\mathbb{Q}$

Proving that $[0,1]\subset\mathbb{R}$ is compact you make use of completeness and this is a fundamental step in order to characterize compact subsets of $\mathbb{R}$. Trying to state an analogous ...
13
votes
9answers
1k views

Why did we define the concept of continuity originally, and why it is defined the way it is?

The concept of continuity is a very important idea in topology. Though I am using it all the time, but indeed I don't know what is the original purpose for us to define this concept. And I also don't ...
13
votes
1answer
323 views

$\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus \{0\}$ are not homeomorphic

Let $k$ be an algebraic closed field. Why $\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus\{0\}$ (for $n>1$) are not homeomorphic with respect to the Zariski topology?
12
votes
4answers
8k views

Cauchy sequence is convergent iff it has a convergent subsequence

Prove that if $\left ( x_{n} \right )$ is a Cauchy sequence in a metric space X then $\left ( x_{n} \right )$ is convergent if and only if $\left ( x_{n} \right )$ has a convergent subsequence. Note: ...
12
votes
1answer
316 views

On different definitions of neighbourhood.

I am going through the basics of topology, mainly to refresh them. I had taken a course some years ago but never used topology actively. So I am reading Munkres's Topology. I have noticed that he ...
12
votes
3answers
295 views

Looking For a Neat Proof of the Fact that the Grassmannian Manifold is Hausdorff

$\newcommand{\R}{\mathbf R}$ Let $V$ be an $n$-dimensional vector space and $k$ be an integer less than $n$. A $k$-frame in $V$ is an injective linear map $T:\R^k\to V$. Let the set of all the $k$-...
12
votes
2answers
318 views

Is every suborder of $\mathbb{R}$ homeomorphic to some subspace of $\mathbb{R}$?

Let $X \subset \mathbb{R}$ and give $X$ its order topology. (When) is it true that $X$ is homeomorphic to some subspace of $Y \subset \mathbb{R}$? Example: Let $X = [0,1) \cup \{2\} \cup (3,4]$, ...
11
votes
1answer
286 views

Product of spaces is a manifold with boundary. What can be said about the spaces themselves?

Suppose I have two topological spaces $X,Y$ and I know that $X\times Y$ is homeomorphic to a manifold with boundary. Can I conclude that $X$ and $Y$ are manifolds (maybe with boundary)? If not, ...
11
votes
3answers
744 views

What's wrong with this definition of continuity?

Consider this definitions: A function $f:X \to Y$ is continuous at $x\in X$ iff for any open neighborhood $V_{f(x)}$ of $f(x)$ there is an open neighborhood $U_{x}$ of $x$ that gets mapped by $f$ ...
11
votes
1answer
287 views

Every path has a simple “subpath”

I've been thinking about this for a while, and can't seem to find any way to do it despite the statement itself seeming obvious. The problem is: Let $f:[0,1] \to \mathbb{R}^n$ be a continuous map,...
11
votes
3answers
777 views

variant on Sierpinski carpet: rescue the tablecloth!

I was playing around with Sierpinski carpets (see pretty GPU-produced picture here), and came up with a variation that I have been unable to find mentioned elsewhere. I'm wondering if anyone can tell ...
11
votes
2answers
291 views

Do all manifolds have a densely defined chart?

Let $M$ be a smooth connected manifold. Is it always possible to find a connected dense open subset $U$ of $M$ which is diffeomorphic to an open subset of R$^n$? If we don't require $U$ to be ...
11
votes
2answers
410 views

A set which is neither meagre nor comeagre in any interval.

As the title suggests, I'm interested in a subset of the real line which is neither meagre nor comeagre in any interval. Does anyone have an example? Added. See the comments for some discussion ...
10
votes
1answer
1k views

Is every countable dense subset of $\mathbb R$ ambiently homeomorphic to $\mathbb Q$

Let $S$ be a countable dense subset of $\mathbb R$. Must there exist a homeomorphism $f: \mathbb R \rightarrow \mathbb R$ such that $f(S) = \mathbb Q$? More weakly, must $S$ be homeomorphic to $\...
10
votes
2answers
369 views

No Smooth Onto Map from Circle to Torus

My professor was lecturing today and he made this statement which I was unable to verify. (I worded it nicer) There is no map which is both smooth and onto from $S^1$ to $S^1$$\times$ $S^1$. When ...
10
votes
1answer
283 views

Is this a sufficient condition for two spaces to be homeomorphic; proof or counter example please.

Let $X$ and $Y$ be topological spaces. let $f: X \to Y$ and $g: Y \to X$. Assume that both $f$ and $g$ are continuous bijections. Can we say that $X$ and $Y$ are homeomorphic? If not are there ...
10
votes
1answer
140 views

Does a map between topologies determine a map between sets?

Let $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ be Hausdorff spaces. Consider a function \begin{equation*} \phi:\mathcal{B}\rightarrow \mathcal{A} \end{equation*} which preserves inclusion, arbitrary ...
10
votes
3answers
6k views

Union of closure of sets is the closure of the union: true for finite, false for infinite unions

Let $A_i$ be a subset of a metric space for each $i\in \mathbb{N}$. Let $B_n := \bigcup_{i=1}^n A_i$. Prove (for any) $n \in \mathbb{N}$ that $\overline{B_n} = \bigcup_{i=1}^n \overline{A_i}$. ...
10
votes
1answer
311 views

Orbit space of a free, proper G-action principal bundle

Let $G$ be a topological group and let $r \colon E \times G \to E$ be a continuous right-action on a topological space $X$. If $p\colon E \to B$ is a continuous map into a topological space $B$ such ...
10
votes
3answers
2k views

Is a discrete set inside a compact space necessarily finite?

Is it true that if $A$ is discrete as a subspace of $X$, and $X \;$ is compact, then $A$ is finite? If this doesn't hold, then does it hold for $X\;$ manifold?
10
votes
1answer
295 views

In the Sorgenfrey plane, is an open disc homeomorphic to an open square?

In the sorgenfrey plane $\mathbb{R}_l^2$, the subspace $$X=\{(x,y):x^2+y^2\leq 1\}$$ is not homeomorphic to the subspace $$Y=\{(x,y):|x|\leq 1,|y|\leq 1\},$$ because there is only one isolated point ...
9
votes
1answer
252 views

A Prime $\mathcal P$-filter is contained in a unique $\mathcal P$-ultrafilter?

Some backround: Let $\mathcal P$ be a class of subsets of a topological space such that if $P_1$ and $P_2$ are sets from $\mathcal P$ then $P_1\cap P_2$ and $P_1\cup P_2$ belong to $\mathcal P$. A $\...
9
votes
1answer
936 views

Big Rudin 1.40: Open Set is a countable union of closed disks?

Reading through Big Rudin, I have come across the following statement in the proof of Theorem 1.40: Let $S \subset \mathbb{C}$ be a closed set [in the topology induced by the complex modulus]. ...
9
votes
2answers
182 views

If a measure only assumes values 0 or 1, is it a Dirac's delta?

Let $\mu$ be a probability measure on a metric space $M$ (with the Borel $\sigma$-algebra). If $\mu(A)\in \{0,1\}$ for all measurable set $A\subset M$, then: Is it true that $\mu$ is a Dirac ...
9
votes
2answers
533 views

weak* separable question

(In another question Nate Eldredge said I should ask this.) Let $X$ be a Banach space, $X^\ast$ the dual space, and $B_{X^\ast}$ the unit ball of $X^\ast$. In the weak* topology for $X^\ast$, does ...
9
votes
5answers
14k views

Example of neither open nor closed set

I need a very simple example of a set of Real numbers ( if there is any ) that is neither closed nor open. And also, a very short and simple explanation of why it is neither closed nor open. Thank you!...
9
votes
1answer
411 views

Clopen subsets of a compact metric space

I am aked to show that in a compact metric space we can find at most countably many subsets which are both: open and close. I would be grateful for your help.
9
votes
1answer
746 views

Closedness of sets under linear transformation

Let $Y$ be a closed subset of $\mathbb{R}^m$ (in fact $Y$ is convex and compact, but I think the extra assumptions are irrelevant). Let $A \in \mathbb{R}^{n \times n}$ be a non-singular matrix (so $A^{...
9
votes
2answers
969 views

“Area” of the topologist's sine curve

Consider the topologist's sine curve: $$ f(x) = \sin\left(1 \over x \right), x \neq 0 $$ The graph of this function resembles a space-filling curve near $x=0$. It is not a space filling curve, ...
9
votes
2answers
605 views

Does the Euler characteristic of a manifold depend upon the field of coefficients?

Define the Euler characteristic of a space $X$ to be $$\chi(X)= \sum_i \dim H_i(X, \mathbb Q)$$ This is obviously not necessarily well-defined for an arbitrary space $X$, so let $X$ be a manifold (...
8
votes
1answer
763 views

Is a connected sum of manifolds uniquely defined?

It is a standard excercise in differential geometry to prove that a connected sum $M\#N$ of two smooth manifolds $M,N$ of the same dimension is uniquely defined (under some assumptions regarding ...
8
votes
0answers
539 views

Questions related to intersections of open sets and Baire spaces

EDIT: I have reposted this question on MathOverflow. (The version posted there is more concise, with some details omitted. I have also added a question about pseudobases with similar property.) MO ...
8
votes
1answer
179 views

Is there a domain in $\mathbb{R}^3$ with finite non-trivial $\pi_1$ but $H_1=0$?

The exterior of the Alexander Horned Sphere has $H_1=0$ but $\pi_1\neq 0$, in fact, $\pi_1$ is infinite. (See Hatcher p.171-172). Is there an example of a domain (connected open set) in $\mathbb{R}^3$ ...
8
votes
2answers
172 views

The topology on $X/{\sim}\times X/{\sim}$ is not induced by $\pi\times\pi$.

I'm looking for an example of a topological space $X$ together with an equivalence relation $\sim$ where the product topology on $X/{\sim}\times X/{\sim}$ is not induced by $\pi\times\pi$ as a final ...
8
votes
2answers
248 views

Is there a nonabelian topological group operation on the reals?

Inspired by A binary operation, closed over the reals, that is associative, but not commutative. That question asks for a noncommutative semigroup operation on $\Bbb R$, for which right projection is ...