analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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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
Given a space , a subspace of , and a concrete point in , is a limit point of if can be approximated by the contents of .
There are several variations on this idea, and the term ‘limit point’ itself is ambiguous (sometimes meaning Definition , sometimes Definition .
The classical definitions apply when is a topological space. Then may be thought of as a subset of (the underlying set of) , and as an element. In order to apply the definitions in constructive mathematics, there needs to be an inequality relation on the points of ; in classical mathematics, this is taken to be the denial inequality, as usual. (We need not assume that is an apartness relation nor any compatibility between and the topology, at least for the definitions; although it's quite possible that some classical theorems will require such assumptions.)
For the most general definitions, let be a collection of cardinal numbers. (We might want to have some closure properties akin to those of an arity class, but the definition there is not quite what we want.) Recall that a -ary indexed subset of is a function such that the cardinality of belongs to the class ; a point is in (as an indexed subset) if belongs to the range of (as a function), and is out of if is inequal () to every point in .
The point is a -adherent point of the subspace if, for each neighbourhood of , for each -ary indexed subset of the intersection , there is an element of that is out of . Slightly more strongly, is a -accumulation point (or -cluster point) of if, for each neighbourhood of , for each -ary indexed subset of , there is an element of that is out of . (Alternatively, take to be a punctured neighborhood, but that won't work constructively in general.)
Every -accumulation point is a -adherent point; the converse holds if every satisfies (and then one usually says ‘accumulation’ rather than ‘adherent’). Also, if , then every -(adherent/accumulation) point is a -(adherent/accumulation) point.
It immediately follows that the following classical special cases are in order of increasing strength:
Using :
The point is an adherent point of the subspace if, for every neighbourhood of , the intersection is inhabited (nonempty).
Using again:
The point is an accumulation point of the subspace if, for every punctured neighbourhood of , the intersection is inhabited.
Using :
The point is an -accumulation point (or -accumulation point) of the subspace if, for every neighbourhood of , the intersection is infinite.
Using :
The point is a condensation point of the subspace if, for every neighbourhood of , the intersection is uncountable.
The subspace is closed iff every adherent point of belongs to and iff every accumulation point of belongs to . (Thus one may say that is closed iff every limit point of belongs to without ambiguity.)
More generally, the closure of is the set of all adherent points of . This justifies using ‘limit point’ to mean an adherent point: the adherent points of are precisely those that are limits of nets of points in . Classically (using excluded middle, or more generally if has decidable equality), the closure of is the union of and its set of accumulation points.
The set of accumulation points of is also called the derived set? of , denoted . The study of derived sets is of great historical importance in Georg Cantor's development of set theory, even though closure sets are more important in modern mathematics. Note that while , no similar relationship holds between and , , etc; one can even continue this into transfinite ordinal numbers (possibly their earliest application).
Classically, a point in that is not an accumulation point of is precisely an isolated point of . (Constructively, each of these is stronger than the negation of the other, but the two conditions may be taken to be antitheses.)
A justification for the terminology ‘limit point’ for an accumulation point is that the concept of limit of a function approaching a point really only makes sense approaching an accumulation point. (This is for essentially the same reason that every function is continuous at an isolated point.) Indeed, every answer whatsoever satisfies the naive definition of if is an isolated point of (because the improper filter converges everywhere).
Last revised on June 27, 2020 at 06:54:50. See the history of this page for a list of all contributions to it.