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
Eventuality filters are the key translation between filters and nets in analysis and topology. Specifically:
Given a net $n$ (or in particular a sequence) in a set $X$, the eventuality filter of $n$ is a proper filter on $X$;
Every proper filter on $X$ is the eventuality filter of some net in $X$;
Two nets are equivalent for purposes of convergence (meaning precisely that they are subnets of each other in the sense of Årnes & Andenæs) if and only if their eventuality filters are equal.
This last point is not so much a result as the definition of the subnet relation (or at least of its symmetrisation, the relation of equivalence of nets). One still needs to check that every use of nets in analysis and topology (or other fields) actually respects this notion of equivalence of nets, if one wishes to convert nets to filters.
Let $X$ be a set, let $D$ be a directed set, and let $n$ be a function from $D$ to $X$, so that $n$ is a net in $X$. Given a subset $A$ of $X$, $n$ is eventually in $A$ if, for some $i$ in $D$, for each $j \geq i$ in $D$, $n_j \in A$. The collection $F_n$ of all those subsets $A$ such that $n$ is eventually in $A$ is a proper filter on $X$, called the eventuality filter of $n$.
In symbols,
where $\ess \forall\, j$ is read ‘for essentially each $j$’ or ‘eventually for each $j$’ and means $\exists\, i,\; \forall j \geq i$. In the case where $D$ is the set of natural numbers directed by $\leq$, so that $n$ is an infinite sequence ($\omega$-sequence) in $X$, then $\ess \forall\, j$ may be read as ‘for all but finitely many $j$’.
Given a filter $F$ on $X$, let $D$ be the binary relation $\in_X$ restricted to $F$, viewed as a subset of the cartesian product $X \times F$, ordered so that $(y, A) \leq (z, B)$ means simply that $B \subseteq A$. Let $n\colon D \to X$ map $(y, A)$ simply to $y$. Then $D$ is directed (so that $n$ is a net in $X$) iff $F$ is proper; and in that case, the eventuality filter of the net $n$ is $F$.
It is possible to define $D$ as ${{\in_X}|_F} \times \mathbb{N}$ (instead of as ${\in_X}|_F$ as done above) so that the direction on $D$ is a partial order, if one really wants to. (Then $(y, A, i) \leq (z, B, j)$ means that $B \subseteq A$, $i \leq j$, and $y = z$ if $i = j$.)
Last revised on May 29, 2022 at 16:17:55. See the history of this page for a list of all contributions to it.