The tag has no wiki summary.

learn more… | top users | synonyms

2
votes
1answer
219 views

Anti-compactness

Let $(X,\tau)$ be a topological space such that $\tau$ is a proper superset of $\{\emptyset, X\}$. We call an open cover $\mathcal{U}$ of $(X,\tau)$ proper if $X\notin \mathcal{U}$. Moreover we say ...
0
votes
0answers
43 views

Baire sets in locally compact Hausdorff spaces

I posed this on 14 Dec. at http://math.stackexchange.com/questions/1067751/baire-sets-in-locally-compact-hausdorff-spaces, but there has been no response: (This is a follow-up to ...
1
vote
1answer
162 views

Relative Compactness vs Way Below in Locally Compact Hausdorff Spaces

Let $Y$ be a subset of a locally compact Hausdorff topological space $X$ and consider the following properties. $\overline{Y}$ is compact. Every open cover of $X$ has a finite subcover of $Y$. ...
4
votes
1answer
168 views

Totally disconnected subspaces

This question is motivated by this one, where no simple solution within ZFC seems to exist. Let me ask a weaker question then. Suppose that $K$ is a compact, Hausdorff, non-metrizable space. Does it ...
1
vote
2answers
125 views

Why the intersection of a scott open (or \w the relatively compactness property) filter on a topology of a sober (and 2nd countable) space is compact?

Definitions and notations. Let $\mathcal{P}(X)$ the power set of $X$. Let $\tau_X\subseteq\mathcal{P}(X)$ a topology on X. We call $A$ irreducible if every time $A=B\cup C$ with $B,C$ closed set ...
12
votes
1answer
259 views

A generalization of the Arhangelskii Theorem

Arhangeleskii's Theorem states the following For any Hausdorff topological space $X$, $$ |X|\leq2^{\chi(X)L(X)} $$ where $\chi(X)$ is the character of $X$ and $L(X)$ is the Lindelöf degree of ...
2
votes
3answers
233 views

Compact, densely ordered spaces

During my work with order preserving homeomorphisms, I got interested in the double arrow space and, subsequently, in the lexicographic square. I would really like to find examples of spaces like ...
2
votes
1answer
135 views

A compactness result: if $f_n(u_n) \rightharpoonup w$ in $L^2(0,T;L^2)$, then $f_n(u_n) \to w$ in $L^2(s,T;H^{-1})$ for all $s > 0$

Let $f_n \to f$ on compact subsets of the real line (these are functions defined on the real line) satisfying some conditions: $f$ has linear growth (but is nonlinear function) and is continuous and ...
5
votes
2answers
307 views

Beyond Cantor's Teepee

From Counterexamples in Topology by Steen and Seebach (2nd edition) example 129 page 145 we have an example of connected and totally path-disconnected space. It is defined as follow: Fix $p= ...
2
votes
0answers
112 views

Hilbert Schmidt Operators and the Conditional Expectation Operator

Consider the function $\text{E}_W: L_2(\mathbb{R},P_X) \mapsto L_2(\mathbb{R},P_W)$ where $P_X$ and $P_W$ are two different probability measures. They are related in such a way that if $f_X$, $f_W$ ...
1
vote
0answers
103 views

If an upper semicontinuous multivalued map is compact on a set, is it compact on the boundary as well?

I have stumbled upon the following problem during my research: Let $X$ and $Y$ be Banach spaces, $K\subset X$ nonempty, $F:\overline{K}\rightarrow 2^{Y}$ an upper semicontinuous multivalued map with ...
10
votes
2answers
433 views

Does there exist a supercompactness theorem?

Large cardinals such as weakly compact cardinals, measurable cardinals, strongly compact cardinals, and extendible cardinals all can be characterized in terms of a certain compactness theorem of ...
3
votes
1answer
168 views

Fréchet-Kolmogorov compactness Theorem for Lp spaces on manifolds

Suppose I have a family of functions $\mathcal{F} \subseteq L^2(\mathcal{M}, P)$ where $\mathcal{M}$ is a compact manifold, and $P$ is a probability distribution on $\mathcal{M}$. Is there an ...
5
votes
1answer
265 views

Locally finite compact groups

I assume all tolpological groups here to be Hausdorff. A group is called locally finite if every finitely generated subgroup is finite. What can be said about a locally finite compact group? Must it ...
1
vote
3answers
194 views

Axiomatization of locally compact Hausdorff spaces via compact subspaces

The usual axiomatization of a topological space (in the sense of Bourbaki) goes by declaring certain subsets as being open and such that a few axioms are fulfilled by the family of open subsets. It ...
2
votes
0answers
106 views

Weak relative compactness in $L^1_{loc}$.

In my work I stumbled upon a proposition (without proof, alas), which I can't really prove. Suppose we have a family of functions $\left\{\phi_\epsilon (t,x,v)\right\}_{\epsilon\in(0,1]}$, and $M(v)$ ...
4
votes
3answers
285 views

Can the intersection of the boundaries of compact and convex sets be a single element?

Let $H_1,H_2,\dots,H_n$ be compact and convex sets in $\mathbb{R}^n$ such that $\bigcap_{j=1}^n H_j$ has non-empty interior and for each $i=1,2,\dots,n$ there exist at least one element $x \in H_i$ ...
17
votes
0answers
477 views

Can closed compacts in a topological group behave “paradoxically” with respect to unions, intersections, and one-sided translations?

Consider two closed compacts $A$ and $B$ in a topological group $\Gamma$. Let $A'$ be a left translation of $A$ and $B'$ a left translation of $B$: $A' = aA$, $B' = bB$. Suppose it is known that ...
3
votes
0answers
114 views

trace-class embeddings

There is a classical theorem of Riesz-Kolmogorov that characterizes compact embedding in $L^p$-spaces of some subspace of them. A generalization to arbitrary metric spaces has been recently obtained ...
0
votes
1answer
192 views

Can an accumulation point be an eigenvalue?

For an discrete (separable) infinite-dimensional Hilbert Space with a compact operator, 0 is always an accumulation point (https://www.math.ucdavis.edu/~hunter/book/ch9.pdf). Does this mean its part ...
2
votes
2answers
154 views

Are compact sets in a Banach lattice order bounded?

Given a compact subset $A$ of a Banach lattice $E$, is the following true? There exist $u,v\in E$ so that $u\leq a\leq v$ for all $a\in A$. This is true in case $E=C(X)$, $X$ compact, with the ...
2
votes
1answer
167 views

A question on countably compact space

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 star countable) $|X|=\aleph_1$ ...
3
votes
3answers
365 views

Compactness of sigma-algebra for the $L^1$ metrics

Consider a probability space $(X,F,\mu)$, and the quotient $G$ of the sigma-algebra $F$ by its null sets. Endow $G$ with the metric $d(A,B) = \mu(A \triangle B)$. Is $(G,d)$ a compact metric space? ...