Lagmental Vicfred

Product Topology Is the Smallest Topology Making Projections Continuous by Vicfred

tags: mathematics, general topology, compactness, connectedness

Finite intersections of inverse images under coordinate projections form a basis for a product. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Objects and 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.

$$ \pi_i:\prod_{j\in J}X_j\to X_i $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ \mathcal B=\left\{\prod_{j\in J}U_j:U_j\text{ open},\ U_j=X_j\text{ for all but finitely many }j\right\} $$

Push the symbols

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ f:Y\to\prod_jX_j\ \text{continuous}\Longleftrightarrow\pi_j\circ f:Y\to X_j\ \text{continuous for every }j $$

Structural reading

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.

$$ \begin{aligned} \mathsf{D}\;&:\quad \pi_i:\prod_{j\in J}X_j\to X_i,\\[5pt] \mathsf{C}\;&:\quad \mathcal B=\left\{\prod_{j\in J}U_j:U_j\text{ open},\ U_j=X_j\text{ for all but finitely many }j\right\}. \end{aligned} $$

A hypothesis worth keeping

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \mathcal B=\left\{\prod_{j\in J}U_j:U_j\text{ open},\ U_j=X_j\text{ for all but finitely many }j\right\} \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Fri 20 February 2009. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.