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) loop in a space $X$ is a map (a morphism in an appropriate category of spaces, such as a continuous function between topological spaces) to $X$ from the circle $S^1 = \mathbb{R}/\mathbb{Z}$. Hence a continuous path whose endpoints coincide.
A loop at $a$ is a loop $f$ such that $f(k) = a$ for any (hence every) integer $k$. An unparametrised loop is an equivalence class of loops, such that $f$ and $g$ are equivalent if there is an increasing automorphism $\phi$ of $S^1$ such that $g = f \circ \phi$. An unoriented loop is an equivalence class of loops such that $f$ is equivalent to $(x \mapsto f(-x))$. A Moore loop has domain $\mathbb{R}/n \mathbb{Z}$ for some natural number (or possibly any real number) $n$. All of these variations can be combined, of course. (A Moore loop at $a$ has $f(k n) = a$ instead of $f(k) = a$. Also, a Moore loop for $n = 0$ is simply a point, so possibly there is a better way to define this to avoid making this exception. Finally, there is not much difference between unparametrised loops and unparametrised Moore loops, since we may interpret $(t \mapsto n t)$ as a reparametrisation $\phi$.)
In graph theory, a loop is an edge whose endpoints are the same vertex. Actually, this is a special case of the above, if we interpret $S^1$ as the graph with $1$ vertex and $1$ edge; in this way, the other variations become meaningful. In this context, a Moore loop is called a cycle. (However, as the only directed graph automorphism of $S^1$ is the identity, parametrisation is trivial for directed graphs and equivalent to orientation for undirected graphs.)
Every loop may be interpreted as a path. Sometimes a loop, say at $a$, is defined to be a path from $a$ to $a$. However, this is correct only in certain contexts. In graph theory, it's incorrect, but only because of terminological conventions; the idea is sound. In continuous spaces, it is also correct. However, in smooth spaces, it is not correct, since the derivatives at the endpoints should also agree; the same holds in many other more structured contexts.
Given two Moore loops $f$ and $g$ at $a$, the concatenation of $f$ and $g$ is a Moore loop $f ; g$ or $g \circ f$ at $a$. If the domain of $f$ is $\mathbb{R}/m \mathbb{Z}$ and the domain of $g$ is $\mathbb{R}/n \mathbb{Z}$, then the domain of $f ; g$ is $\mathbb{R}/(m+n) \mathbb{Z}$, and
In this way, we get a monoid of Moore loops in $X$ at $a$, with concatenation as multiplication. This monoid may called the Moore loop monoid?.
Often we are more interested in a quotient monoid of the Moore loop monoid. If we use unparametrised loops (in which case we may use loops with domain $S^1$ if we wish), then we get the unparametrised loop monoid?. If $X$ is a smooth space, then we may additionally identify loops related through a thin homotopy to get the loop group. Finally, if $X$ is a continuous space and we identify loops related through any (basepoint-preserving) homotopy, then we get the fundamental group of $X$.
See looping and delooping for more.
Last revised on July 2, 2017 at 06:42:07. See the history of this page for a list of all contributions to it.