What is Wittgenstein claiming when he says that "each thing can be the case or not the case while everything else remains the same"?

Question

In Tractatus 1.21 Wittgenstein writes

 Each item can be the case or not the case while everything else remains the same. [1]

I'm looking for commentary on this point, since it seems out of place given other statements in the Tractatus, and also with just plain logic (what about the entry of the item into other combinations? Etc). This is possibly due to my unfamiliarity with W's thinking at this basic level (I'm a mathematician who's worked on category-theoretic logic), but it might also present some train of thought that never quite got clarified, for instance arising out of something from the Prototractatus that was later altered.

Also, leaving aside the issue about W's whole repudiation of Tractatus, did he ever address this specific point later?

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[1] The original, with Prototractatus numbering included, is

1.2 [78(9)] Die Welt zerfällt in Tatsachen.

1.21 [78(10)] Eines kann der Fall sein oder nicht der Fall sein und alles Übrige gleichbleiben.

Answer 1

The title "Tractatus logico-philosophicus" indicates the strong affinity to logic.

In my opinion Proposition 1 means that Wittgenstein considers the world - or better a logical model of the world - as the set of all true propositions. In the most simple case a compound proposition is a conjunction of component propositions. Wittgenstein assumes a decomposition into independent component propositions. A priori, each of these component propositions can be true or false, independent from each other.

Apparently, if the compound proposition is true, i.e. it refers to a fact, then also each component proposition must be true, i.e. refers to a fact. And vice versa.