Definition talk:Irreducible Component

Irreducible Component should be a subset

A component is usually a subset, not a subspace. I assume this definition is a mistake, since it is indeed wrongly regarded as a subset e.g. in Irreducible Component is Closed and Closure of Irreducible Subspace is Irreducible.

You say "usually" and not "always", therefore defining an "irreducible component" as a subspace is not necessarily wrong. Since a subset under the subspace topology is always a subspace I can't see what the problem is. --prime mover (talk) 17:46, 30 August 2022 (UTC)

If this is just a mistake, it is better to correct it. The problem is the difficulty of using this. It makes this definition useless. Look at this:

Let $T = \struct {S, \tau}$ be a topological space.

A subspace $Y \subset T$

What is $Y$ now? As $T$ is a pair, is $Y$ also a pair? It is too difficult to formulate something using this definition, correctly.

For example, if we write $Y=\struct {S_Y,\tau_Y}$ and want to consider a closure of $S_Y$ as a subset of $S$, we need to explain a lot of unessential things. --Usagiop (talk) 18:45, 30 August 2022 (UTC)

First of all, we need to define the inclusion $\struct {S_Y,\tau_Y} \subseteq \struct {S,\tau}$.--Usagiop (talk) 18:52, 30 August 2022 (UTC)

Is there any advantage in referencing the concept of "component"?

One of the problems I have with this definition is that it references "irreducible subset" which then references "irreducible space" which then goes onto a page with a ridiculous 7 definitions, only 2 of which are sourced. (I hate that.) It would probably be worth including some descriptive words into this definition that provide at least an intuitive idea as to exactly what it means to be an "irreducible component" rather than rely upon a string of links. --prime mover (talk) 21:44, 30 August 2022 (UTC)

I assume you essentially mean Definition:Irreducible Subset is a duplication of Definition:Irreducible Space. Maybe the former can be cleverly merged into the latter. The problem I forced is the ambiguity of the subjects. Because a topological subspace of $T$ is a pair, it is formally not a subset of $T$. The issue seems ignored or tolerated in Definition:Compact Space. I am trying to make this definition more precise. --Usagiop (talk) 22:45, 30 August 2022 (UTC)

For example, I cannot simply say that $S = A_1 \cup A_2$ for irreducible components $A_1$ and $A_2$, if irreducible components are defined as topological subspaces. I have no idea how to say this, effectively.--Usagiop (talk) 23:07, 30 August 2022 (UTC)

So are we at the stage of saying: delete everything and start again from scratch? --prime mover (talk) 05:42, 31 August 2022 (UTC)

Slightly inaccurate cross-references should be tolerated with hope of future collaborative improvement. Only mathematically wrong expressions must be quickly corrected.--Usagiop (talk) 16:28, 31 August 2022 (UTC)

Can we correct this? Or does someone know the source for this definition?

I improved the definition in a backward compatible way.--Usagiop (talk) 19:29, 30 August 2022 (UTC)

By the way I am going to add the related pages:

Definition:Irreducible Subset

Irreducible Component Decomposition of Closed Set in Noetherian Space

--Usagiop (talk) 17:12, 30 August 2022 (UTC)

Barto posted up loads of excrement because he was very clever and knew everything and was therefore exempt from having to bother with either rigour or accuracy. I would have gone through and cleaned everything up but I decided I would rather get a job as a toilet cleaner. --prime mover (talk) 17:44, 30 August 2022 (UTC)