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; ...
1
vote
0answers
15 views
Basically disconnectedness $\Rightarrow$ Extremally disconnectedness
There exists Basically disconnected space which is not extremally disconnected, The one-point Lindelöfization of an uncountable discrete space is a such spaces. But with what conditions basically ...
3
votes
1answer
28 views
Definition of convergence in $C^\infty(\Omega)$
Apparently, some of you will count it a silly question; but I am really not convinced or understand, the way they define convergence and then topology as a consequence of convergence.
$\Omega$ is ...
2
votes
1answer
35 views
A separable, regular space which has cardinality of the continuum but is not first countable?
Actually the title says it all. Is there such a topological space which is separable, regular, has cardinality of the continuum but is not first countable?
If so, is there also an example of a ...
6
votes
3answers
78 views
Question about a proof that $\mathbb{Q}$ is dense in $\mathbb{R}$
This is from Ross's elementary analysis book. The statement is if $a,b \in \mathbb{R}$ such that $a<b$ then there exists a rational $r \in \mathbb{Q}$ such that $a<r<b$.
I don't understand ...
0
votes
3answers
47 views
Is this piecewise-defined function on $\mathbb{R}^2$ continuous at $(0,0)$? What about differentiable?
Is this function is a differentiable function, a continuous function at the point $(0,0)$?
How to show that ?
$$
f(x,y)=\begin{cases}
\frac{x^{3}y+xy^{3}}{x^{2}+y^{4}} &\text{if }(x,y)\neq ...
8
votes
1answer
45 views
$X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ with subspace topology Then
$X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ with subspace topology. Then
$0$ is an isolated point
$(-2,0]$ is an open set
$0$ is a limit point of the subset $\left\{{1\over ...
3
votes
1answer
36 views
Question on Urysohn's lemma
$A_1=\text{ Closed Unit Disk}$, $A_2=\{(1,y):y\in\mathbb{R}\}$, $A_3=\{(0,2)\}$.
Then there always exists a real-valued continuous function on $\mathbb{R}^2$ such that
$f(x)=a_j$ for $x\in A_j$, ...
1
vote
1answer
51 views
Dimensions analysis in Differential equation
Differential equation of solitary wave oscillons is defined by,
$$ \Delta S -S +S^3=0 $$
How can we write this equation as,
\begin{equation}
\langle(\vec{\nabla}S)^2\rangle+\langle S^2\rangle-\langle ...
0
votes
1answer
61 views
Properties of Continuous Functions
Prove that there is no continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for $c \in \mathbb{R}$ the equation $f(x)=c$ has exactly two solutions.
This is what I have so far.
Proof by ...
1
vote
1answer
82 views
Compact metric space group $Iso(X,d)$ is also compact
Could you tell me how to prove that if metric space $(X,d)$ is compact, then the group $Iso(X,d)$ is also compact?
The group $Iso(X,d)$ is considered with topology determined by a metric $\rho$ on ...
1
vote
1answer
60 views
Prove that the dimension of the tangent space $T_x(X)$ of a k-dimensional manifold is k
In section 2, page 9 of Guillemin and Pollack's book $\textit{Differential Topology}$, he gave a proof that the dimension of the tangent space $T_x(X)$ is equal to the dimension of the manifold $X$. ...
1
vote
1answer
50 views
All closed balls are compact each isometry is bijective
Let $(X,d)$ be a metric space in which all closed balls are compact and such that for any two points $x,y \in X$ there exists a function $u \in Iso(X,d)$ such that $u(x)=y$.
Prove that then each ...
0
votes
1answer
60 views
How many Borel conjectures are there
The following may be referred to as Borel conjecture:
Every strong measure zero set of reals is countable.
On the other hand Wikipedia refers to the following as the Borel conjecture:
Let $M$ and ...
2
votes
1answer
38 views
Proof that the interior of any union of closed sets with empty interior in a compact Hausdorff space is empty
The question is pretty much in the title, I need to show that given $X$ is a compact Hausdorff space and $\left\{ A_n\right\}_{n=1}^\infty$ is a collection of closed subsets of $X$ each with empty ...
2
votes
0answers
29 views
Uniformly continuous function - modulus of continuity
Give an example of a uniformly continuous function $f: (X,d) \rightarrow (Y,\rho)$ for which there doesn't exists a modulus of continuity $\omega: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ such that:
...
12
votes
3answers
134 views
Is a torus a subset of $\mathbb{R}^3$ or $\mathbb{R}^4$?
Defining a torus $T$ as $S^1 \times S^1$, it should follow that $T \subseteq \mathbb{R}^4$. But you can also think of a torus as a bagel, which means it's a subset of $\mathbb{R}^3$. Can anyone ...
2
votes
2answers
47 views
Confusion about Lemma 13.2 in Munkres' topology (property which implies that a collection is a basis for a topology)
Lemma 13.2 and its proof confuse me.
$X$ is a topological space and $C$ is a collection of open sets of $X$ satisfying a property. A specific topology is not mentioned in the lemma. In the proof, ...
6
votes
1answer
91 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 ...
1
vote
1answer
31 views
The algebraic possibilities of the (topological) procedure of the compactification of a space
If $X$ is locally compact $K$-vector space, then $X\cup \{\infty\}$ is via the Alexandroff-compactification a compact space.
But this purely topological procedure tells me nothing about the algebraic ...
2
votes
1answer
45 views
Function spaces and transitive group actions
Note: this question is really a subquestion of this one, but I decided to ask it separately since it seems it might be attacked first.
Let $B$ be a topological space and $G$ a topological group ...
2
votes
3answers
52 views
How easily to check: $f$ is a closed map?
Assume that $f:\mathbb R\to\mathbb R$ is a function where $\mathbb R$ is the real number and the usual topology is defined on $\mathbb R$.
I have two questions:
1. Let $C$ be closed. Then what is ...
1
vote
3answers
57 views
The representation of closed set
We already know that closed set is the complement of open set.
But I want to know the representation of closed set without the fact above in the usual topology.
For instance, the integers ...
2
votes
0answers
54 views
Weakly closed subsets of $C(K)$
Given a compact Hausdorff space $K$, let us endow $C(K)$ with the Banach-space weak topology. Is there any handy description of weakly closed subsets of $C(K)$? Are subsets of $C(K)$ which are ...
5
votes
2answers
39 views
Regular $T_2$ space which is not completely regular.
Theorem 10. of
Pontryagin's Topological Groups says that:
Every Hausdorff topological group is completely regular.
But is there exists a Regular $T_2$ space which is not completely regular?
1
vote
2answers
47 views
Application of Urysohn's lemma
I am working on the following hw problem: If we have that $X$ is a compact Hausdorff space, with $\{U_\alpha\}_{\alpha\in A}$, then we can find a finite number of continous functions $f_1,...,f_k$, ...
8
votes
0answers
46 views
Are there more embeddings $U(2) \hookrightarrow SO(4)$?
It is easy to prove that $SO(4)$ acts transitively and freely on $S^2$ with fiber $U(2)$. Therefore, we can identify each point of $S^2$ with a particular embedding $U(2) \hookrightarrow SO(4)$.
My ...
1
vote
2answers
44 views
Topology - Dunce Cap Homotopy Equivalent to $S^2$
So I'm trying to find two spaces with isomorphic homology groups but where the spaces aren't homotopy equivalent.
From my work so far, taking the Dunce Cap as a triangle with the edges identified as ...
6
votes
2answers
68 views
Two terms that I want to understand: weakest topology and jointly continuous (in the following context).
I was reading an article online, please help me to understand the following lines (in bold letters). -
Topological structure:
If (V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric and ...
8
votes
1answer
82 views
Topological manifolds (dimension)
I am taking an introductory course to topology and the professor defined a topological manifold of dimension $n$ if it is hausdorff and if for every point $x$ there exists an open set $U$ around $x$ ...
17
votes
0answers
134 views
Does $X\times S^1\cong Y\times S^1$ imply that $X\times\mathbb R\cong Y\times\mathbb R$?
This question came up in a recent video series of lectures by Mike Freedman available through Max Planck Institut's website. He proves the "difficult" converse direction, that $X\times \mathbb R\cong ...
-1
votes
0answers
28 views
Continuity of quotient map
$f:X\to X/\mathord{\sim}$ is continuous for any space $X$ and equivalence relation $\sim$, where $f$ is defined to be $f(x)=[x]$, and $[x]$ is the equivalence class of $x$.
How can I prove this?
2
votes
1answer
53 views
What's the fundamental group of $E^2\setminus Q^2$
Here $E^2$ is the two-dimensional Euclid space and $Q$ is the set of all rational numbers. Regard $E^2\setminus Q^2$ as a subspace of $E^2$. So what's its fundamental group and how to represent it? I ...
4
votes
2answers
89 views
Why doesn't the Weierstrass approximation theorem imply that every continuous function can be written as a power series?
I hope that my question in the title is well formulated.
I am a little bit confused with the next exercise from a book:
Argue that there exists functions $f \in C[0, \frac{1}{2}]$ that cannot be ...
1
vote
1answer
45 views
Exercise on Lebesgue measure ( Treatise of Analysis Vol2 by Dieudonné)
Someone challenge me to bring the solution from anywhere! So I have posted here and see, I am optimist because this website is excellent and its members are so helpful.
Let me start with this ...
2
votes
1answer
56 views
Is closure of a semigroup again a semigroup?
Let $S$ be a compact left-topological semi-group (meaning, $S$ is both a semi-group and a compact Hausdorff topological space, and the map $x \mapsto x y$ is continuous for any fixed $y$, but the map ...
0
votes
1answer
19 views
Maximal square covering
Let X be a shape in 2-dimensional space.
Define a square covering of X as a set of axis-aligned squares, whose union exactly equals X.
Note that some shapes don't have a finite square covering, for ...
6
votes
1answer
72 views
Topology for convergent sequences
Let $(X,\tau)$ be a topological space, and consider the family $\mathcal{F}$ of the topologies over $X$ such that the convergent sequences for each $\gamma \in \mathcal{F}$ are the same as the ...
4
votes
2answers
49 views
Is any compact metric totally disconnected space homeomorphic to a compact subspace of a Cantor space?
Every compact metric totally disconnected perfect space is homeomorphic to a Cantor space.
Is every compact metric totally disconnected space homeomorphic to a compact subspace of a Cantor space?
...
-1
votes
0answers
50 views
Product topology question
Consider $\mathbb R$ with the usual topology and let
$S^1=\{(x,y)\in \mathbb R^2: x^2+y^2 = 1\} $.
a) $( \mathbb R^2, \Gamma_{p})$, where $\Gamma_p$ is the product topology, coincides with ...
3
votes
2answers
66 views
Baire's theorem from a point of view of measure theory
According to Baire's theorem, for each countable collection of open dense subsets of $[0,1]$, their intersection $A$ is dense. Are we able to say something about the Lebegue's measure of $A$? Must it ...
2
votes
0answers
31 views
Topology of the Segre product vs. the product topology
In general, the product topology on two (quasiprojective) varieties is not the same as the topology of the product variety given by the Segre embedding. This is something I've often seen asserted is ...
1
vote
0answers
71 views
Compactness property
Let $\Omega \subset X$, X: Banach space. Given $\varepsilon \ge 0$, we define the set of $\varepsilon-normals$ to $\Omega$ at $\bar{x}$$\in \Omega$ by:$\widehat N_\varepsilon(\bar ...
2
votes
1answer
31 views
The restriction of a covering map on the connected component of its definition domain
Suppose $p:Y\to X$ is a covering map, $X,Y$ are manifolds and $X$ is connected. If $Z$ is a connected component of $Y$, I wonder if the restriction of $p$ on $Z$ is also a covering map? If not, what ...
0
votes
1answer
23 views
Is the ascending union of contractible spaces contractible
Let $\{Y_i\}_{i \in \mathbb{N}}$ be a collection of subspaces of $X$ such that each $Y_i$ is contractible and $Y_{i} \subseteq Y_{i+1}$. Is $\bigcup_{i\in \mathbb{N}}Y_i \subseteq X$ also ...
3
votes
1answer
61 views
Infinite products of a (finite) group
So I'm having a little trouble understanding the concept of infinite (cartesian) products of a group -- specifically, my notes (and, of course, homework questions) have concepts of, say ...
1
vote
2answers
60 views
Continuous Function + open set
A mapping $T$ of a metric space $X$ into a metric space $Y$ is continuous iff the inverse image of any open subset $Y$ is open subset of $X$.
Proof:
(a)Suppose that $T$ is continuous. Let $S \subset ...
2
votes
2answers
36 views
Topological extension property
Let $X$ and $Y$ be topological spaces. We say that the extension property holds if, whenver $S$ is a closed subset of $X$ and $f:S\rightarrow Y$ is continuous, $f$ can be extended to a continuous ...
3
votes
2answers
47 views
$X$ topological space. $A$ open $A \cap Y = \emptyset \ \ \Longrightarrow A \cap \bar{Y} = \emptyset$?
I know this is an easy question, but I cannot demonstrate it properly.
Suppose by contradiction that $A \cap \bar{Y} \neq \emptyset$. Then $\exists \ x \in A \cap \bar{Y}$.
I need help formalizing ...
1
vote
0answers
19 views
Calculating Topological Genus
How would I calculate the topology of a sphere, with a smaller sphere inside removed.
I know if I drill through to get to the hole, then we are back to being a sphere, and if I drill out to the other ...
4
votes
1answer
27 views
Characterization for compact sets in $\mathbb{R} $ with the topology generated by rays of the form $\left(-\infty,a\right) $
I'm trying to find a sufficient and necessary condition for a subset to be compact in $\mathbb{R} $ when the topology is generated by the basis $\left\{ \left(-\infty,a\right)\,|\, ...
