Examples/classes:
Types
Related concepts:
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
Writhe is a measure of how much a knot or link writhes around itself. As a 1-dimensional line cannot actually twist, this is not a knot invariant but is an invariant of framed knots (and links). Since a link diagram can be given a natural framing (the blackboard framing), it is possible to compute the writhe of a specific diagram. One place where this is used very neatly is to convert the Kauffman bracket?, which is an invariant of framed links, into the Jones polynomial, being an invariant of ordinary links.
Recall that a framed link can be thought of as a link together with a normal direction along each component, which we call the framing direction.
The writhe of a framed link is the linking number of the link with its infinitesimal displacement in the framing direction.
For an oriented link diagram, the writhe is defined using the orientation of the crossings.
The writhe of an oriented link diagram is defined to be the sum of the orientations of its crossings.
Last revised on June 28, 2017 at 18:52:32. See the history of this page for a list of all contributions to it.