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
In topology, a (parametrised, oriented) path in a space $X$ is a map (a morphism in an appropriate category of spaces, such as a continuous function between topological space) to $X$ from the topological interval $\mathbb{I} = [0,1]$.
A path from $a$ to $b$ is a path $f$ such that $f(0) = a$ and $f(1) = b$.
An unparametrised path is an equivalence class of paths, such that $f$ and $g$ are equivalent if there is an increasing automorphism $\phi$ of $\mathbb{I}$ such that $g = f \circ \phi$. An unoriented path is an equivalence class of paths such that $f$ is equivalent to $(x \mapsto f(1 - x))$.
If $P$ is a path, then its reverse path^{1}, denoted $\overline{P}$, is defined to be the composite $P \circ ( t\mapsto 1-t )$. The operation $P\mapsto\overline{P}$ is called path reversal.
A Moore path is defined like a path, except for having another domain: replace $[0,1]$ with the interval $[0,n]$ for some natural number (or, more commonly, any non-negative real number) $n$. All of these variations can be combined, of course. (For unoriented paths, one usually says ‘between $a$ and $b$’ instead of ‘from $a$ to $b$’. Also, a Moore path from $a$ to $b$ has $f(n) = b$ instead of $f(1) = b$. Finally, there is not much difference between unparametrised paths and unparametrised Moore paths, since we may interpret $(t \mapsto n t)$ as a reparametrisation $\phi$.)
In graph theory, a path is a list of edges, each of which ends where the next begins. Actually, this is a special case of the above, if we use Moore paths and interpret $[0,n]$ as the linear graph with $n + 1$ vertices and $n$ edges; in this way, the other variations become meaningful. (However, as the only directed graph automorphism of $[0,n]$ is the identity, parametrisation is trivial for directed graphs and equivalent to orientation for undirected graphs. Note that a non-Moore path is simply an edge, one of the fundamental ingredients of a graph.)
Given a Moore path $f$ from $a$ to $b$ and a Moore path $g$ from $b$ to $c$, the concatenation of $f$ and $g$ is a Moore path $f ; g$ or $g \circ f$ from $a$ to $c$. If the domain of $f$ is $[0,m]$ and the domain of $g$ is $[0,n]$, then the domain of $f ; g$ is $[0,m+n]$, and
In this way, we get a (strict) category whose objects are points in $X$ and whose morphisms are Moore paths in $X$, with concatenation as composition. This category is called the Moore path category.
Often we are more interested in a quotient category of the Moore path category. If we use unparametrised paths (in which case we may use paths with domain $\mathbb{I}$ if we wish), then we get the unparametrised path category. If $X$ is a smooth space, then we may additionally identify paths related through a thin homotopy to get the path groupoid. Finally, if $X$ is a continuous space and we identify paths related through any (endpoint-preserving) homotopy, then we get the fundamental groupoid of $X$.
In graph theory, the Moore path category is known as the free category on the graph.
Tammo tom Dieck, Algebraic Topology, European Mathematical Society, 2008
Cf. e.g. Introduction to Topology – 2, or also Section 2.1; beware that that reference, (0) like many others, uses the term “inverse path”, even though the operation of concatenation of paths does not in and of itself yield a strict groupoid, in which $\overline{P}$ would be an inverse, and (1) that it uses $a$ and $b$ for the endpoints of the interval, not the endpoints of the paths in the space $X$, and (2) that it uses $P^-$ instead of $\overline{P}$, which however is less suited for notational iterating (compare $\overline{\overline{P}}=P$ with $(P^-)^-=P$), and that (3) the 2008 edition has a typo: “ $w(1-t)$ ” in loc. cit., when inverse path gets defined, should be $u(1-t)$. ↩
Last revised on September 21, 2021 at 09:07:26. See the history of this page for a list of all contributions to it.