Lagmental Vicfred

A Basis Generates Open Sets by Unions by Vicfred

tags: mathematics, general topology, compactness, connectedness

A basis covers the space and refines every intersection around each common point. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Notation

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.

$$ \mathcal B\subseteq2^X $$

I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).

$$ \tau(\mathcal B)=\left\{\bigcup_{\alpha}B_\alpha:B_\alpha\in\mathcal B\right\} $$

Stress the formula

A worked instance is useful here because it exposes every index that the compressed statement hides.

$$ x\in B_1\cap B_2\Longrightarrow\exists B_3\in\mathcal B:\quad x\in B_3\subseteq B_1\cap B_2 $$

Interpretation

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad \mathcal B\subseteq2^X,\\[5pt] \mathsf{C}\;&:\quad \tau(\mathcal B)=\left\{\bigcup_{\alpha}B_\alpha:B_\alpha\in\mathcal B\right\}. \end{aligned} $$

Limit of the argument

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \tau(\mathcal B)=\left\{\bigcup_{\alpha}B_\alpha:B_\alpha\in\mathcal B\right\} \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Sun 17 October 2021. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.