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
graphics grabbed from Lawson 03
The Möbius strip is the topological space obtained from the “open strip”, hence the square, by gluing two of its opposite sites but after applying half of a full rotation to one of them.
As a topological space, the Möbius strip is the quotient topological space obtained from the square by the equivalence relation which identifies two of the opposite sides, but with opposite orientation
Regarded a vector bundle over the circle, the Möbius strip is the tautological line bundle over the 1-dimensional real projective space .
For more discussion of the topological vector bundle structure see this example and this prop.
Regarded as a manifold, the Möbius strip is among the simplest examples of a manifold that is not orientable.
Regarded as a real vector bundle over the circle, the Möbius strip is among the simplest examples of a non-trivial vector bundle.
Named after August Möbius.
Last revised on November 22, 2020 at 19:54:02. See the history of this page for a list of all contributions to it.