Lagmental Vicfred

Compactness Converts Open Covers into Finite Data by Vicfred

Last updated: Mon 30 August 2021

tags: mathematics, general topology, compactness, connectedness

A compact space admits a finite subcover from every open cover. This is a compact note, but the quantifiers and hypotheses stay on the page.

Definitions first

A topology \(\tau\subseteq2^X\) specifies which subsets of \(X\) are open. Continuity is defined by inverse images, so it composes without requiring coordinates or distances.

$$ X=\bigcup_{\alpha\in A}U_\alpha $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ X\ \text{compact}\Longrightarrow\exists\alpha_1,\ldots,\alpha_n:\quad X=\bigcup_{i=1}^{n}U_{\alpha_i} $$

A small case in full

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ X\ \text{compact},\ Y\ \text{Hausdorff},\ f:X\to Y\ \text{continuous}\Longrightarrow f(X)\ \text{compact and }f\text{ closed onto }f(X) $$

The reusable statement

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad X=\bigcup_{\alpha\in A}U_\alpha,\\[5pt] \mathsf{C}\;&:\quad X\ \text{compact}\Longrightarrow\exists\alpha_1,\ldots,\alpha_n:\quad X=\bigcup_{i=1}^{n}U_{\alpha_i}. \end{aligned} $$

A nearby false statement

Compact, connected, path-connected, and Hausdorff are independent properties in general spaces. Metric-space intuition supplies implications only with extra hypotheses.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] X\ \text{compact}\Longrightarrow\exists\alpha_1,\ldots,\alpha_n:\quad X=\bigcup_{i=1}^{n}U_{\alpha_i} \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Mon 17 April 2017. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.