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In the proof : If $ A$ is separable, then $ \Delta(A)$ satisfies the second axiom of countability. ($ \Delta(A)$ the set of all complex homomorphisms of $ A$ .) I have found they says “It is easy to see that the unit ball of the adjoint space of a separable Banach space is second-countableRead more

Let $ K/F$ be a field extension and $ \alpha \in K$ is algebraic over the field $ F.$ Now suppose $ \alpha$ is separable over $ F.$ Then how can I show that $ F(\alpha)/F$ is a separable extension, i.e., an arbitrary element of $ F(\alpha)$ is a root of a separable polynomial overRead more

Introduction This is really a parsing question. But the context is that given the right side of first order ode, I want to automatically determine the class of the RHS. It can be either separable, or linear. If it is none of these, then I can do something else. For learning, I am writing aRead more

A space $ X$ is a $ \sigma$ -space if $ X$ has a $ \sigma$ -discrete network. Let $ X$ be a Lindelof, perfectly normal, $ \sigma$ -space. Must $ X$ be separable? Thanks very much.Read more

A topological space has calibre $ \aleph_1$ if for every uncountable sequence $ \langle U_\alpha\mid\alpha\lt\aleph_1\rangle$ of nonempty open sets $ U_\alpha\subset X$ , there is an uncountable subfamily $ \Lambda\subset\aleph_1$ with $ \bigcap_{\alpha\in\Lambda}U_\alpha\neq\emptyset$ . Is there a calibre $ \aleph_1$ Moore space which is not separable? (Under CH, the answer is no, since every firstRead more