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
Examples/classes:
Types
Related concepts:
A homotopy between two maps $f,g \;\colon\; X \to Y$ may be thought of as a continuous path in the mapping space $Map(X,Y)$. Similarly, a path in the subspace $Emb(X,Y) \hookrightarrow Map(X,Y)$ of embeddings of $X$ into $Y$, is an isotopy.
Since, for arbitrary topological spaces, the mapping space $Map(X,Y)$ is not always well-behaved, defining a homotopy as a left homotopy using a cylinder object, hence as a map $H \colon X \times I \to Y$ is the rigorous way of saying “$H$ is a path from $f$ to $g$ in the mapping space.”. Similarly, the proper definition of an isotopy as a map $H \colon X \times I \to Y$ with the property that $H(-,t) \colon X \to Y$ is always an embedding, is the rigorous way of saying “$H$ is a path from $f$ to $g$ in the embedding space.”.
In some settings, even this is not strong enough. In the notion of an ambient isotopy, one requires the path to extend to a path in the space of homeomorphisms of the ambient space.
Let $X$ and $Y$ be two topological spaces. Let $f,g \colon X \to Y$ be two embeddings of $X$ in to $Y$. An isotopy from $f$ to $g$ is a continuous map $H \colon X \times [0,1] \to Y$ with the following properties:
Two maps for which there exists an isotopy are said to be isotopic.
Let $X$ and $Y$ be two topological spaces. Let $f,g \colon X \to Y$ be two embeddings of $X$ in to $Y$. An ambient isotopy from $f$ to $g$ is a continuous map $H \colon Y \times [0,1] \to Y$ with the following properties:
The first two conditions on an isotopy $H$ imply that $H$ is a homotopy from $f$ to $g$. Therefore the condition of being isotopic is stronger than that of being homotopic. As the third condition applies level-wise, the proof that homotopy is an equivalence relation carries over to show that the same is true of isotopy.
Isotopy is used where one wishes to study deformations of an object inside some ambient space that do not change the object itself. An extremely important example of this is the theory of knots and links where, to prevent unknottings and unlinkings, there have to be some restrictions on the allowed movements. These are usually encoded in terms of isotopies. It is unfortunately true, however, that a naive use of isotopy leads to strange results. If you use continuous isotopies then any two continuous embeddings of the circle into $S^3$ are isotopic, (basically since you just pull the knot tighter and tighter, (so the knotted bit gets smaller and smaller) and at the end you just put the ‘unknot’). One way to handle this is to demand the isotopy to be piecewise linear? (or smooth), another is to work explicitly with ambient isotopy.
One of the beauties of isotopy of knots is that it can be realised very simply at the level of knot diagrams. Two knots are isotopic if their respective knot diagrams can be related using Reidemeister moves. (This is a formal theorem, but will be given elsewhere after some more development.)
General discussion includes
For isotopy in knot theory see
R. H Crowell and R.H. Fox, Introduction to Knot Theory, Springer, Graduate Texts 57, 1963.
N.D. Gilbert and Tim Porter, Knots and Surfaces, Oxford U.P., 1994.
Specifically for smooth isotopy:
Last revised on January 31, 2024 at 12:28:43. See the history of this page for a list of all contributions to it.