Topologically

A topological space is Hausdorff if and only if it is both preregular (i.e. topologically distinguishable points are separated by neighbourhoods) and Kolmogorov (i.e. distinct points are topologically distinguishable).

Research on insulators with topologically protected surface conductivity – in short: topological insulators – is currently in vogue.

A C*-algebra is a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties: * A is a topologically closed set in the norm topology of operators.

Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces.

Because of this, they are not necessarily topologically equivalent to the sphere as Platonic solids are, and in particular the Euler relation : does not always hold.

Conversely, the Lindenstrauss–Tzafriri theorem asserts that if every closed subspace of a Banach space is complemented, then the Banach space is isomorphic (topologically) to a Hilbert space.

Every polygon is topologically self-dual (it has the same number of vertices as edges, and these are switched by duality), but will not in general be geometrically self-dual (up to rigid motion, for instance).

Indeed, given a list of nodes, and for each node a list of references to its children, one cannot tell if this structure is a tree or not without analyzing its global structure and that it is in fact topologically a tree, as defined below.

More generally, Euler's formula applies to any polyhedron whose faces are simple polygons that form a surface topologically equivalent to a sphere, regardless of its convexity.

Symmetry mutations This polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.

There exist also two-dimensional polymers which are composed of topologically planar repeat units.

Therefore the topology on a topologically finitely-generated profinite group is uniquely determined by its algebraic structure.

The smallest scalenohedron has 8 faces and is topologically identical to the regular octahedron.

The two embedded tori are homeomorphic, but not isotopic : They are topologically equivalent, but their embeddings are not.

This is because 1) two-dimensional shapes (surfaces) are the lowest-dimensional shapes for which nonorientability is possible, and 2) the Möbius strip is the only surface that is topologically a subspace of every non-orientable surface.

This is because there are morphisms of schemes which are topologically open immersions but which are not scheme-theoretic open immersions.

Thus, compact Riemann surfaces are characterized topologically simply by their genus.

Topological duality We can distort a dual polyhedron such that it can no longer be obtained by reciprocating the original in any sphere; in this case we can say that the two polyhedra are still topologically or abstractly dual.

Topologically sorting is easy in contrast to counting the number of topological sortings.

Topologically, the family P(θ) is just a line (because each line in P(θ) intersects the line L(θ) in just one point).