Articles of compactness

Characterization of compactness in weak* topology

Let $ X $ be Banach space, and $X^*$ its dual. A set $ F \subset X ^ * $ is weakly-* compact if and only if $ F $ is closed in the weak* topology and is bounded in norm. How does one prove this (well-known) fact? Notes This characterization of compactness in weak* […]

Understanding one point compactification

I just read about one point compactification and i am having some difficulty in grasping the concept. Does one point compactification mean that we are simply adding a point to a non compact space to make it compact. For example, my book says that $S^n$ is the one point compactification of $\mathbb R^n$, i don’t […]

one point compactification of discrete space

Problem: What is the one point compactification $X^*$ of a discrete space $X$. In the case of $X$ being finite, $X$ itself is compact so the one point compactification would be merely $X$ $\bigcup$ {$\infty$}. Now in the case of $X$ being infinite I need to consider two cases. When $X$ is countably infinite, I […]

Compact subspace of a Banach space .

The following statement doesn’t make sense to me, can someone justify it to me ? If $K$ is a compact subset of a Banach space $Y$ then there exists for $\epsilon > 0 $ a finite dimensional subspace $Y’$ of $Y$ such that $d(x, Y’) < \epsilon $ for every $x \in K $ , […]

Prove that the set of extreme points of a compact convex set is not empty.

The Krein–Milman theorem states that if $S$ is convex and compact in a locally convex space, then $S$ is the closed convex hull of its extreme points. In particular, such a set has extreme points. Is there a “simple argument” that the extreme points is not empty that avoids going through the full Krein-Milman theorem?

Why can't we express in first logic order even number of nodes in graph

I can show using compactness theorem (or even EF Games) that even number of nodes in graphs is not axiomatizable. However, I don’t understand some thing: We can express cardinality of universum, so we should be able to express number of nodes with infinity set of formulas (formula for each even number expressing cardinality of […]

A direct proof that a compact metric space is sequentially compact

I am looking for a direct proof (not by contradiction) that a compact metric space is sequentially compact, ie constructing a converging subsequence from any sequence. Thanks

characterisation of compactness in the space of all convergent sequences

I go through a proof of the following. Let $(\ell_1,d)$ be the metric space of all sequences $x = (\xi_i)_{i \in \mathbb{N}}$ with $\sum_{i=1}^{\infty} |\xi_i| < \infty$ and the metric $$ d(x,y) = \sum_{i=1}^{\infty} |\xi_i – \eta_i|, \qquad x = (\xi_i), y = (\eta_i). $$ Theorem: A subset $M$ of $l_1$ is totally bounded (pre-compact) […]

Compact $G_\delta$ subsets of locally compact Hausdorff spaces

Suppose $X$ is a locally compact Hausdorff space and $F$ is a closed subset thereof. Then of course $F$ is also locally compact and Hausdorff. Let $K$ be a subset of $F$, and suppose that $K$ is a compact $G_\delta$ with respect to $F$. Clearly $K$ is compact as a subset of $X$. Must it […]

For a Compact Metric Space $T: X\to X$ has unique fixed point

given: $(X,d)$ is a compact metric space $T:X\to X$ is such that $d(T(x),T(y))<d(x,y)\ \forall x,y\in X$ with $x\neq y$. Prove that T has a unique fixed point. Attempt: I think I can prove Uniqueness: Consider $T(x)=x$ and $T(y)=y,\ x\neq y$ Then, $d(x,y)=d(T(x),T(y))<d(x,y)$ (contradicting original condition). However, I am having trouble executing the proof that the […]