The compactness tag is for questions about compactness and its many variants (e.g. sequential compactness) as well locally compact spaces; compactifications (e.g. one-point, Stone-Cech) and other topics closely related to compactness.
3
votes
2answers
72 views
A question about a continuous function on a compact metric space
suppose that (X,d) is a compact metric space and $f:X\to X$ is a continuous function.
Define $X_1 = f(X), X_2 = f(X_1),...,X_{i+1} = f(X_i),...$ and let $A = \bigcap_{i=1}^\infty X_i$. Is $A \subseteq ...
2
votes
2answers
56 views
Is my proof correct? (minimal distance between compact sets)
I'm working out the following problem form Ahlfors' Complex Analysis text:
"Let $X$ and $Y$ be compact sets in a complete metric space $(S,d)$. Prove that there exist $x \in X,y \in Y$ such that ...
4
votes
1answer
67 views
Poincaré inequality and Rellich Theorem in one dimensional weighted Sobolev space
Consider the weighted Sobolev space $W^{1,2}\big((0,R),r^{N-1}\big)$, $N=2,3,\ldots$ and its subspace $W_0^{1,2}\big((0,R),r^{N-1}\big)$. Anyone knows if the Poincaré inequality is true in this case?
...
1
vote
1answer
37 views
Partition on metric space
Let be $\gamma = \{C_1,\ldots, C_k\}$ a partition of a compact metric space $X$ such that $diam(C_j)<\delta$ for all $j$. Suppose that there exist compact sets $L_i\subset C_i$ for all ...
1
vote
1answer
104 views
SHOW that there are infinitely many equivalence classes of formulas
Let $\mathcal{Q}$ denote the additive group of rational numbers, i.e. the structure $\left<\mathbb{Q}; +; 0\right>$. Let $\mathcal{L}$ be the
language of $\mathcal{Q}$ and let $T$ be the ...
1
vote
1answer
87 views
the diameter of nested compact sequence
Let $E_{i}$ be a nested compact subsequence s.t $\forall i E_i\geqslant r$ for $r>0$ how can we show that this implies that $\bigcap_{i=1}^\infty E_i$ also has parameter bigger than $r$?
0
votes
1answer
27 views
Can a Accumulation Point be an Eigenvalue?
I have a discrete (separable) infinite dimensional Hilbert Space with a compact operator defined on it. So 0 is an accumulation point (some theorem says so). Can 0 also be an eigenvalue? And how would ...
0
votes
1answer
82 views
Proving sequentially compact
How does one prove that a set is sequentially compact? I'm having trouble picturing what it means for every sequence to have a subsequence that converges.
Examples would be greatly appreciated too.
...
16
votes
0answers
139 views
If $S\times\mathbb{R}$ is homeomorphic to $T\times\mathbb{R}$, and $S$ and $T$ are compact, can we conclude that $S$ and $T$ are homeomorphic?
If $S \times \mathbb{R}$ is homeomorphic to $T \times \mathbb{R}$ and $S$ and $T$ are compact, connected manifolds (according to an earlier question if one of them is compact the other one needs to be ...
6
votes
0answers
70 views
An example of a compact multiplicatively unbounded ring
My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
5
votes
0answers
89 views
Is dependent choice necessary to prove every perfect compact Hausdorff space is uncountable?
The answer to Cardinality of a locally compact space without isolated point shows that AC is required to show that if $X$ is a compact Hausdorff space with no isolated points then $|X| \ge ...
4
votes
0answers
54 views
Is $X$ pseudocompact
The following example with a little modified from the handbook of set theoretic topology, Page 574:
Let $\kappa$ be any cardinal for which there exists a family $\{H_\alpha: \alpha < \kappa\}$ ...
4
votes
0answers
244 views
If $f: X \to Y$, when do we have $\beta Y \supset \overline{f(X)} = \beta X$?
Suppose that $X$ and $Y$ are Tychonoff spaces, denote by $\beta X$ and $\beta Y$ their Stone-Čech compactifications and let $f:X\to Y$ be a continuous map.
Using the embedding $Y\hookrightarrow\beta ...
4
votes
0answers
52 views
A question on countably compact space under CH
The question is also posted here.
A regular space $X$ is
star compact (which implies pseudocompact)
with $G_\delta$-diagonal
star countable
first countable
$e(X)\le \aleph_0$ ( in fact it implies ...
4
votes
0answers
61 views
Classes of compact spaces
Given a class $\mathcal{C}$ of compact Hausdorff spaces which is closed under countable products and continuous images. Let $\kappa>\omega$ be a cardinal number. Consider the class ...
3
votes
0answers
41 views
Another question in relation to Tychonoff theorem
Let $X_i$ be compact topological spaces and let $X = \prod_{i \in I}X_i$ and let $\mathscr F$ be ultrafilter on $X$. Define $\mathscr F_i = \{Y \subseteq X_i : \pi_i^{-1}Y \in \mathscr F\}$. Here ...
3
votes
0answers
72 views
What is the point of compactness and how does it look visually?
I just learned what the term compact means, as in compact sets. I know what the definition of compactness is, but I don't get what the real significance of it is. How should I try to think of it in a ...
3
votes
0answers
179 views
Sequential compactness vs. countable compactness
Firstly, I'll give the definitions of sequential compactness and countable compactness.
Sequential compactness: If $X$ is a Hausdorff space and every sequence of points of $X$ has a convergent ...
2
votes
0answers
75 views
For a family of functions $F\subset C(X)$ in the metric space $(C(X),d)$, if $F$ is compact on compact subsets of $X$, then $F$ is compact on $X$
The problem as stated in the title isn't quite correct. Let $X$ be a topological space. What I have is a family of functions $F\subset C(X)$ in the metric space $(C(X),d)$ which on compact subsets ...
2
votes
0answers
83 views
Models of infinite cardinal and compactness
I'm stuck with this problem:
$L$ is first-order language with identity and $L_q$ a language obtained by adding to $L$ the quantifier $Q$. If $P$ is a formula and $x$ a variable, $QxP$ is a formula ...
2
votes
0answers
48 views
Show compactness of an evolution operator
Consider the heat equation
$$
u_{t}=u_{xx},~~~~~u_0(x)=u(0,x)$$
with $u\colon [0,T]\times\mathbb{R}\to\mathbb{R}, (t,x)\mapsto u(t,x)$
and the evolution operator $E(T)$ with $E(T)u_0=u(T,x)$.
1.) ...
2
votes
0answers
102 views
Connectedness of the complement of a compact “small” subset of $\mathbb R^n$
Let $C$ be a compact subset of $\mathbb R^n$ and suppose that for every $\varepsilon >0$ there exists a finite family of open disks $B_i$ s.t. $C \subset \bigcup_{i} B_i$ and $\sum_i r_i \le ...
1
vote
0answers
49 views
If $X$ is a compact space, what toplogical properties can its dense left-separated subspace have?
Any space contains a dense left-separated subspace.
Given any space $X$, form a (possibly transfinite) sequence of points by the following induction. As long as (the range of) the sequence you've ...
1
vote
0answers
77 views
Example of compactness of the set of elements of a convergent sequence.
An example in a book I am reading on general topology demonstrates the concept of compactness by showing that the set $E = \{s_n : n = 0,1,2,3 \ldots \}$ is compact in some topological space S. The ...
0
votes
0answers
39 views
Convex Hull of Precompact Subset is Precompact
I'm trying to prove that, if $K$ is a precompact (I've also heard the phrase totally bounded used for this) subset of a Banach Space $X$, then its convex hull is also precompact.
I've come across a ...
0
votes
0answers
31 views
Exhaustive sequence
For each of the following domains $D$, find an exhaustive sequence $K_n$ of nested compact subsets of $D$.
(a) $D$ is an open disc
(b) $D$ is an open annulus
(c) $D$ is the plane with $k$ pairwise ...
0
votes
0answers
29 views
conditions ensuring $\lim\sup$ of a increasing sets is compact
Let $(E_n)_{n>0}$ be a nondecreasing sequence of compact sets in $\mathbb{R}^d$ (that is $E_n\subseteq E_{n+1}$).
Under which conditions $\lim\sup E_n$ is compact?,
Note that the monotonicity of ...
0
votes
0answers
77 views
What's wrong with my proof that a continuous function is uniformly continuous?
I was trying to prove that any continuous function from a compact metric space to any other metric space is uniformly continuous.
I proved it as follows:
Let $f\colon X\to Y$ be continuous and ...
0
votes
0answers
40 views
Convergence of left Cauchy nets and compactness
Is every left Cauchy net in a compact set convergent? Why?
If no, is every left Cauchy net in a compact group convergent? Why?
0
votes
0answers
36 views
The existence of an attractor for a finite family of continuous transformations
Let $(X, \rho)$ be a complete metric space and let $S_1,...,S_N: X\rightarrow X$ be continous functions.
A nonempty compact set $C$ is said to be an attractor for the family $\{S_1,...,S_n\}$ if ...
0
votes
0answers
85 views
Convolutions, Compact Support and the Divergence Theorem
Ok, first off, this is a long question, so apologies for that. My LateX isn't up to par, so I've coded what I can and I've linked the rest.
I've just proved this, and it probably leads on to the bit ...
0
votes
0answers
145 views
Connected component
Read on topological spaces I found two doubts on the following result:
Let $T$ a compact and Hausdorff topological space and $C_t$ the connected component for $t\in T$.
a) Why if $T$ is compact ...
