Simply connected space in the context of "Path (topology)"

In physical cosmology, the shape of the universe refers to both its local and global geometry. Local geometry is defined primarily by its curvature, while the global geometry is characterised by its topology (which itself is constrained by curvature). General relativity explains how spatial curvature (local geometry) is constrained by gravity. The global topology of the universe cannot be deduced from measurements of curvature inferred from observations within the family of homogeneous general relativistic models alone, due to the existence of locally indistinguishable spaces with varying global topological characteristics. For example; a multiply connected space like a 3 torus has everywhere zero curvature but is finite in extent, whereas a flat simply connected space is infinite in extent (such as Euclidean space).

Current observational evidence (WMAP, BOOMERanG, and Planck for example) imply that the observable universe is spatially flat to within a 0.4% margin of error of the curvature density parameter with an unknown global topology. It is currently unknown whether the universe is simply connected like euclidean space or multiply connected like a torus.

The Friedmann–Lemaître–Robertson–Walker metric (FLRW; /ˈfriːdmən ləˈmɛtrə ... /) is a metric that describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected, but not necessarily simply connected. The general form of the metric follows from the geometric properties of homogeneity and isotropy. Depending on geographical or historical preferences, the set of the four scientists – Alexander Friedmann, Georges Lemaître, Howard P. Robertson, and Arthur Geoffrey Walker – is variously grouped as Friedmann, Friedmann–Robertson–Walker (FRW), Robertson–Walker (RW), or Friedmann–Lemaître (FL). When combined with Einstein's field equations, the metric gives the Friedmann equation, which has been developed into the Standard Model of modern cosmology and further developed into the Lambda-CDM model.

Cosmic strings are hypothetical 1-dimensional topological defects which may have formed during a symmetry-breaking phase transition in the early universe when the topology of the vacuum manifold associated to this symmetry breaking was not simply connected.

In less formal terms, they are hypothetical long, thin defects in the fabric of space. They might have formed in the early universe during a process where certain symmetries were broken. Their existence was first contemplated by the theoretical physicist Tom Kibble in the 1970s.

In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic).

In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982) as part of his 24 questions, and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.