topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Let Top be a category of topological spaces and an object in (the ‘base’ space). The slice category is called the category of (topological) spaces over (or sometimes simply bundles).
An étale space (or étale map, sometimes étalé space) over is an object in such that is a local homeomorphism: that is, for every , there is an open set such that the image is open in and the restriction of to is a homeomorphism .
The set where is called the stalk of over .
The underlying set of the total space is the union of its stalks (notice that we do not say fiber!). is sometimes refered to as the projection.
The following is adapted from a MathOverflow answer by David Carchedi:
As long as has a small set of topological generators (i.e., as long as isn’t too large to be a topos), there always exists a certain version of an étale space: If is a sheaf on , the slice topos has a canonical étale projection . This map is a local homeomorphism of topoi. This topos with this local homeomorphism is the étale space of . Indeed, we may make this construction for each object , call it , where is the (possibly sheafified) Yoneda embedded object. Then, “sections of over ” are in bijection with elements of . If the Grothendieck site happened to be the canonical site of a topological space, then each slice is equivalent to sheaves on the étale space of that sheaf, and the projection corresponds to the usual one. In particular, corresponds to the inclusion of an open subset. So, this is reduces to the usual construction for spaces. Another example is, if were the small étale site of some scheme , then each is the small étale site of some algebraic space (with no separation conditions) étale over , with corresponding to the étale map from this algebraic space to .
Let be in . The (local) sections of over an open set are the continuous maps such that . It is an elementary but central fact that for an étale map , the images of local sections form a base for the topology of the total space . The topology of is then typically non-Hausdorff.
The set of sections of over is denoted by and may be shown to extend to a functor where is the category of presheaves over . The functor has a left adjoint , whose essential image is the full subcategory of étale spaces over . The essential image of the functor is the category of sheaves over , and this adjunction restricts to an equivalence of categories between and (that is, it is an idempotent adjunction).
If is a sheaf, then one sometimes calls the total space of the étale space the space of the sheaf , having in mind the adjoint equivalence above. (This is also called the sheaf space or the display space (alias étale space, or in French: espace étalé); compare also a display morphism of contexts.) The associated sheaf functor decomposes as , and may be considered as an endofunctor part of an idempotent monad in whose corresponding reflective subcategory is .
(e.g. MacLane-Moerdijk, section II.5, II.6)
Every covering space (even in the more general sense not requiring any connectedness axiom) is étale but not vice versa:
for a covering space the inverse image of some open subset in the base needs to be, by the definition, a disjoint union of homeomorphic open sets in ; however the ‘size’ of the open neighborhoods over various in the same stalk required in the definition of étale space may differ, hence the intersection of their projections does not need to be an open set, if there are infinitely many points in the stalk.
even if the the stalks of the étale space are finite, it need not be locally trivial. For instance the disjoint union of a collection of open subsets of a topological space with the obvious projection is étale, but does not have a typical fiber: the fiber over a given point has cardinality the number of open sets that contain this particular point.
In French, the verb ‘étaler’ means, roughly, to spread out; ‘-er’ becomes ‘-é’ to make a past participle. So an ‘espace étalé’ is a space that has been spread out over . On the other hand, ‘étale’ is a (relatively obscure, distantly related) nautical adjective that can be translated as ‘calm’ or ‘slack’.
To quote from the Wiktionnaire français:
‘étale’ qualifie la mer qui ne monte ni ne descend à la fin du flot ou du jusant (‘flot’ = ‘flow’ and ‘jusant’ = ‘ebb’).
The French pronunciation of “étalé” is three syllables; that of “étale” is two syllables, with the final “e” silent. Many English-speaking mathematicians follow the respective French pronunciations; see for example this nForum thread.
Others use the two-syllable pronunciation (corresponding to French “étale”) for both written forms “étale” and “étalé”. This group included Grothendieck himself, when speaking English. (See McLarty18, from 4:45 to 6:05 and particularly at 5:26.)
There is an interesting stanza from a song of Léo Ferré:
Et que les globules figurent
Une mathématique bleue,
Sur cette mer jamais étale
D’où me remonte peu à peu
Cette mémoire des étoiles.
— (Léo Ferré, La mémoire et la mer)
He also mentions geometry and ‘théorème’ elsewhere in the song.
For generalizations to étale spaces of stacks in groupoids see
Last revised on July 24, 2022 at 07:06:58. See the history of this page for a list of all contributions to it.