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Given some point in a topological space , the union of any collection of connected subsets such that each contains will once again be a connected subset. The connected component of a point in is the union of all connected subsets of that contain ; it is the unique largest (with respect to ) connected subset of that contains .
In two dimensions, a circle is not simply connected, but a disk and a line are. Spaces that are connected but not simply connected are called non-simply connected or multiply connected. A sphere is simply connected because every loop can be contracted (on the surface) to a point. The definition rules out only handle-shaped holes. A sphere (or ...
A space is locally path connected if and only if for all open subsets U, the path components of U are open. [24] Therefore the path components of a locally path connected space give a partition of X into pairwise disjoint open sets. It follows that an open connected subspace of a locally path connected space is necessarily path connected. [25]
In mathematics, a locally simply connected space is a topological space that admits a basis of simply connected sets. [1] [2] Every locally simply connected space is also locally path-connected and locally connected. The Hawaiian earring is not locally simply connected. The circle is an example of a locally simply connected space which is not ...
A topological space is said to be connected if it is not the union of two disjoint nonempty open sets. [2] A set is open if it contains no point lying on its boundary; thus, in an informal, intuitive sense, the fact that a space can be partitioned into disjoint open sets suggests that the boundary between the two sets is not part of the space, and thus splits it into two separate pieces.
An equivalent definition of homotopical connectivity is based on the homotopy groups of the space. A space is n-connected (or n-simple connected) if its first n homotopy groups are trivial. Homotopical connectivity is defined for maps, too. A map is n-connected if it is an isomorphism "up to dimension n, in homotopy".
The Hawaiian earring is not semi-locally simply connected. A simple example of a space that is not semi-locally simply connected is the Hawaiian earring: the union of the circles in the Euclidean plane with centers (1/n, 0) and radii 1/n, for n a natural number. Give this space the subspace topology.
Connected space, a mathematical concept in topology Path-connected space; Simply connected space; Connected ring, a concept from commutative algebra; ConnectEd, a plan to provide high-speed Internet service to nearly all United States schools; Connected (website), The Arts Society (UK) platform to support isolated people during coronavirus pandemic