4
votes
1answer
36 views

Topological property: set-theoretically large subsets of an infinite space are not compact.

Let $X$ be an infinite topological space. Say that $X$ satisfies # if no subset of $X$ of cardinality $|X|$ is compact. So for instance it is clear that no (infinite) compact space satisfies # any ...
1
vote
1answer
38 views

compactness and maximal elements

Let $C$ be a nonempty compact subset of $R^n$, with a certain partial order defined on it. I am trying to prove that $C$ contains a maximal element. My idea is: start with a certain element of $C$. ...
1
vote
1answer
33 views

Approximation by finite sets

I'm reading the book "Topology and Order" by L.Nachbin. In chapter $3$ he speaks about properties of compact Hausdorff spaces. He writes: [A]lthough these spaces may be infinite, they admit ...
1
vote
1answer
56 views

What does “weakly compact” mean when applied to subsets $X \subset Y$?

Let $X$ be a subset of a Banach space $Y$. Please can you give me a definition of what "$X$ is weakly compact" means? I want one which is in terms of sequences and boundedness, as opposed to one with ...
3
votes
0answers
72 views

References for the Čech-Stone compactification of Hyper-Reals?

It seems like $\beta\mathbb R$ has been heavily studied, but I am interested in learning more about $\beta(\mathbb R ^\omega /u)$. $\mathbb R ^\omega /u$ is a proper extension of $\mathbb R$ when ...
1
vote
2answers
77 views

a condition equivalent to compactness in linearly ordered spaces

Does anyone know where can I find a proof to this proposition: A linearly ordered topological space is compact if and only if every bounded subset has an infimum and a supremum. Thank you,
4
votes
0answers
119 views

Locally connected and compact Hausdorff space invariant of continuous mappings

I am looking for a reference (not proof) to the following theorem: If $X$ is a compact and locally connected topological space, Y is a Hausdorff topological space, $f:X\to Y$ is continuous and ...
3
votes
1answer
92 views

The 'compactness cardinal' of a space

I'm looking for references (and a name!) for the following invariant of a topological space $X$: The least (infinite) cardinal $\kappa$ such that any open cover of $X$ has a subcover of cardinality ...
0
votes
0answers
37 views

Stone-Čech compactification [duplicate]

I'm looking for the proof of Stone-Čech compactification for the following Banach algebra $A=C_b(\Omega)$ where $\Omega$ is a completly regular space and $C_b(\Omega)$ is the space of all bounded ...
2
votes
1answer
286 views

Are compacta in a complete infinite dimensional normed space nowhere dense?

Let $X$ be an infinite dimensional Banach space. I want to show that any compact subset $\varnothing\neq A\subset X$ is nowhere dense. I've been able to prove the statement for ...
-1
votes
1answer
142 views

Stone-Cech compactification of the separable Hilbert space

Where can I read about the Stone-Cech compactification of the separable Hilbert space?
2
votes
1answer
205 views

Top cohomology detecting compactness

Could someone point me to a standard reference for the fact that the top cohomology $H^n(M,A)$ of an $n$-dimensional manifold $M$ is non-trivial for local coefficients $A$ if and only if the manifold ...
5
votes
1answer
205 views

Paracompact and Compactly Generated spaces

A couple of days ago, thanks to Strom's excellent book Modern Classical Homotopy Theory, I started reading up on compactly generated spaces, weak Hausdorff spaces and compactly generated weak ...