# The notion of "Closed, Open" regarding Vector Spaces, Topological Spaces and Metric Spaces

When it comes to topologies, my understanding is that we designate elements of that topology as 'open.' In the context of a metric space, a topology is formed by selecting open sets, as defined with respect to a given metric. Moreover, a normed vector space is equipped with an induced metric derived from its norm. This metric allows us to associate it with a metric space and, consequently, with a topology.

In the realm of topology, a closed set is one for which the complement is open. In the context of metric spaces, it refers to the set containing all points to which sequences converge. Now, in regard to normed vector spaces, when discussing Banach spaces, it's often mentioned, as seen in Riesz's lemma, that a certain subspace is 'closed.' I wasn't familiar with a concrete definition of what a 'closed' vector space entails. To the best of my knowledge, this term has been used to describe the property of addition and scalar multiplication remaining closed operations.

Furthermore, are the concepts of 'open' and 'closed' equivalent in these structures? Does dimension play a role? In the case of finite dimensions, all norms are equivalent, and open sets are the same regardless of the induced metrics.

• In Riesz's lemma, "the subspace is closed" means that it's a closed set when you view the whole space as a metric/topological space. This isn't intended to be a property of the subspace as a vector subspace, but rather just as a subset. It is definitely confusing that there's also the concept of being "closed under addition" and "closed under multiplying by scalars"! These are unrelated concepts, and I would expect a mathematician who's being careful, especially when writing about normed spaces, to always say "closed under ..." to refer to these concepts. Commented Oct 20, 2023 at 13:45