locally profinite and profinite group - Mathematics Stack Exchange It is clear that new locally compact totally disconnected groups can be obtained from the consideration of direct products, semidirect products, etc of already known locally compact totally disconnected groups
abstract algebra - Locally Noetherian schemes are quasiseparated . . . $\begingroup$ It is better if you know that every affine open of a locally Noetherian scheme is the spectrum of a Noetherian ring That way you are not stuck with just the given covering $\mathrm{Spec}(A_i)$ $\endgroup$
Local freeness in vector bundles and projective modules A locally free sheaf is only the same as a locally free module over an affine scheme variety There's no finite presentation condition required A trivialization of a vector bundle is a cover of your space by Zariski open sets such that the restriction of your bundle to each open set in the cover is isomorphic to the trivial vector bundle (of
Locally compact metric space - Mathematics Stack Exchange So any incomplete locally compact metric space is a counter-example to "only if" Moreover, as mentioned Tsemo Aristide's answer, any non-compact metric space, even a proper one, has the same topology as some improper metric space A normed space X is proper iff it is locally bounded (iff it is finite-dimensional)
Is connected component open? - Mathematics Stack Exchange There is a theorem that:A space is locally connected iff each connected components of an open set is open But recently I had seen to prove That each connected component is closed Connected Components are Closed Then how can the connected component of an open set be open if it is a locally connected space ?