Canonical projections of the product topology

Stub grade: A*

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Demote once more content is added, and statements are added

Grade: B

This page requires references, it is on a to-do list for being expanded with them.

Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:

At least one would be good

Definition

Let [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath] be an arbitrary family of topological spaces and let [ilmath](\prod_{\alpha\in I}X_\alpha,\mathcal{J})[/ilmath] denote their product, considered with the product topology, then, for each [ilmath]\beta\in I[/ilmath] we get a map:

• [ilmath]\pi_\beta:\prod_{\alpha\in I}X_\alpha\rightarrow X_\beta[/ilmath] given by: [ilmath]\pi_\beta:(x_\alpha)_{\alpha\in I}\mapsto x_\beta[/ilmath]

---

TODO: Add claims, eg continuity and such

---

Sometimes denoted by [ilmath]p_\beta:\prod_{\alpha\in I}X_\alpha\rightarrow X_\beta[/ilmath] instead. We'll use the two interchangeably but will always define them as a canonical projection.

---

TODO: Link to category theory

---

See also

• Canonical injection of the subspace topology
• Canonical injections of the disjoint union topology