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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 topology, a topological space is called simply connected (or 1-connected, or 1-simply connected [1]) if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. Intuitively, this corresponds to a space that has no disjoint parts and no ...
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 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]
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".
In topology and related areas of mathematics a uniformly connected space or Cantor connected space is a uniform space U such that every uniformly continuous function from U to a discrete uniform space is constant. A uniform space U is called uniformly disconnected if it is not uniformly connected.
The property of being a connected space in topology. Homotopical connectivity, a property related to the dimensions of holes in a topological space, and to its homotopy groups. Homological connectivity, a property related to the homology groups of a topological space.