In mathematics, a locally simply connected space is a topological space that admits a basis of simply connected sets.Every locally simply connected space is also locally path-connected and locally connected.The circle is an example of a locally simply connected space which is not simply connected.…

These are just the connected components of M, which are open sets since manifolds are locally-connected.Being locally path connected, a manifold is path-connected if and only if it is connected.It follows that the path-components are the same as the components.…

Many authors impose some connectivity conditions on the spaces X and C in the definition of a covering map.In particular, many authors require both spaces to be path-connected and locally path-connected.This can prove helpful because many theorems hold only if the spaces in question have these properties.…

Similarly, a topological space is said to be if it has a base of path-connected sets.An open subset of a locally path-connected space is connected if and only if it is path-connected.This generalizes the earlier statement about R n and C n, each of which is locally path-connected.…

In particular, they are locally simply connected, locally path connected, and locally connected.

The space X has a universal cover if it is connected, locally path-connected and semi-locally simply connected.The universal cover of the space X can be constructed as a certain space of paths in the space X.…

The (non-extended) long line or ray is not paracompact.It is path-connected, locally path-connected and simply connected but not contractible.It is a one-dimensional topological manifold, with boundary in the case of the closed ray.…

We say that X is locally path connected at x if for every open set V containing x there exists a path connected, open set U with x \in U \subset V, The space X is said to be locally path connected if it is locally path connected at x for all x in X.Since path connected spaces are connected, locally path connected spaces are locally connected.This time the converse does not hold (see example 6 below).…

In the relative topology, a small open subset of is composed of infinitely many almost parallel line segments on the surface of the torus.This means that is not locally path connected.In the group topology, the small open sets are single line segments on the surface of the torus and is locally path connected.…

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.…

The comb space is path connected but not locally path connected.

An open subset of a locally path-connected space is connected if and only if it is path-connected.This generalizes the earlier statement about R n and C n, each of which is locally path-connected.More generally, any topological manifold is locally path-connected.…

Essentially there is an obstruction to the existence of a universal cover which is also a topological group such that the covering map is a morphism: this obstruction lies in the third cohomology group of the group of components of G with coefficients in the fundamental group of G at the identity.If H is a path-connected, locally path-connected, and semilocally simply connected group then it has a universal cover.By the previous construction the universal cover can be made into a topological group with the covering map a continuous homomorphism.…

Every finite space is locally path-connected since the set is a path-connected open neighborhood of x that is contained in every other neighborhood.In other words, this single set forms a local base at x.…

Again suppose is a covering map and C (and therefore also X) is connected and locally path connected.

Note that by the unique lifting property, if f is not the identity and C is path connected, then f has no fixed points.Now suppose is a covering map and C (and therefore also X) is connected and locally path connected.…

Moreover, the path components of the topologist's sine curve C are U, which is open but not closed, and C \setminus U, which is closed but not open.A space is locally path connected if and only if for all open subsets U, the path components of U are open.Therefore the path components of a locally path connected space give a partition of X into pairwise disjoint open sets.…

Examples of hyperconnected spaces include the cofinite topology on any infinite space and the Zariski topology on an algebraic variety.Every hyperconnected space is both connected and locally connected (though not necessarily path-connected or locally path-connected).The continuous image of a hyperconnected space is hyperconnected.…

There are also local versions of these definitions: locally connected spaces and locally path connected spaces.Simply connected spaces are those that, in a certain sense, do not have "holes".…

This generalizes the earlier statement about R n and C n, each of which is locally path-connected.More generally, any topological manifold is locally path-connected.…