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A topological space is said to be disconnected if it is the union of two disjoint non-empty open sets. Otherwise, is said to be connected.A subset of a topological space is said to be connected if it is connected under its subspace topology.
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".
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.
A point of a connected space is called a non-cut point [1] of if {} is connected. Note that these two notions only make sense if the space is connected to start with. Also, for a space to have a cut point, the space must have at least three points, because removing a point from a space with one or two elements always leaves a connected space.