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)

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)