What are presentable categories? - Mathematics Stack Exchange I will speak about $\mathbf{Cat}$ as an ordinary locally small category There is no significant difference between the the various notions because filtered colimits in $\mathbf{Cat}$ preserve equivalences, and so a category is finitely presentable in the bicategorical sense if and only if it is equivalent to one that is finitely presentable in the enriched sense, and because the 2-categorical
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 ?
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
calculus - Is locally linear an appropriate description of a . . . I think there is a big difference between "locally linear" and other uses of "locally" For instance, "locally compact", "locally connected" means, (essentially, or implies) that every point has a neighborhood which IS compact connected "Locally Euclidean" means that every point has a neighborhood which IS $\mathbb
Locally compact metric space - Mathematics Stack Exchange So any incomplete locally compact metric space is a counter-example to "only if" Moreover, as mentioned Tsemo Aristide's answer, any non-compact metric space, even a proper one, has the same topology as some improper metric space A normed space X is proper iff it is locally bounded (iff it is finite-dimensional)
general topology - Definition of a locally Euclidean space . . . Let's say we want a general notion of "$(X,\tau)$ is locally homeomorphic to $(Y,\sigma)$ " In general, $(Y,\sigma)$ may not have the same "self-similarity" property of $\mathbb{R}^n$ which makes the two definitions of "locally Euclidean" equivalent So we get two inequivalent candidates for a general notion of "locally homeomorphic" here
general topology - What is the relationship between completeness and . . . Every locally compact metric space will be an open subset of its completion, and every G$_\delta$ subset of a complete metric space is completely metrizable Every locally compact completely regular space is Čech-complete ( i e , is a G$_\delta$ subset of its Stone-Čech (or any other) compactification), and a metric space is completely