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Subset and subspace synonymous sometimes?

https://en.wikipedia.org/wiki/Dense_set

A is dense in X if and only if the only closed subset of X containing A is X itself.

However, also

https://math.stackexchange.com/a/2019863/248602

Particularly,

is contained in some closed proper subspace of X, and therefore cannot be dense.

Or is it because a subspace is always a subset. However, that a subset may not necessarily be a subspace?

mavavilj
  • 7,270
  • There is a big difference between subset and subspace. Being a subset of $X$ simply means that you only contain points (or functions, or objects, etc) that lie inside $X$, and nothing more. Being a subspace usually carries additional meaning; for instance, a vector subspace requires you to be a subset, plus contain the zero element of X, be closed under addition, etc. – TomGrubb Jun 06 '18 at 16:19
  • Any subset can be equipped with the subspace topology, which makes it a subspace, but for some purposes (such as defining "dense subset") all we need is a subset. –  Jun 06 '18 at 16:20
  • @Bungo So subspace is a stronger form of subset? – mavavilj Jun 06 '18 at 16:20
  • Subspace is a topological space, subset is a set. Those are two different things – Jakobian Jun 06 '18 at 16:21
  • A subspace is a subset along with a particular topology on that subset. –  Jun 06 '18 at 16:21

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