Definition A topological space X is said to be locally connected if, given any point x of X, and given any open set U in X with x ∈ U, there exists some connected open set V in X such that x ∈ V and V ⊂ U.

Definition A topological space X is said to be locally path-connected if,  given any point x of X, and given any open set U in X with x ∈ U, there exists some path-connected open set V in X such that x ∈ V and V ⊂ U.

Definition A topological space X is said to be locally simply-connected if,  given any point x of X, and given any open set U in X with x ∈ U, there exists some simply-connected open set V in X such that x ∈ V and V ⊂ U.

Definition A topological space X is said to be contractible if the identity map of X is homotopic to a constant map that sends the whole of X to a single point of X.

Definition A topological space X is said to be locally contractible if, given any point x of X, and given any open set U in X with x ∈ U, there exists some contractible open set V in X such that x ∈ V and V ⊂ U.

Definition A topological space X is said to be locally Euclidean of dimension n if, given any point x of X, there exists some open set V such that x ∈ V and V is homeomorphic to some open set in n-dimensional Euclidean space Rⁿ.

Note that every locally Euclidean topological space is locally contractible,  every locally contractible topological space is locally simply-connected, every locally simply-connected topological space is locally path-connected, and every locally path-connected topological space is locally connected.

Remark A connected topological space need not be locally connected, alocally connected topological space need not be connected. A standard example is the comb space. This space is the subset of the plane R²consisting of the line segment joining (0,0) to (1, 0), the line segment joining (0,0) to  (0, 1), and the line segments joining (1/n,0) to (1/n,1) for each positive integer n. This space is contractible, and thus simply-connected, path -connectedand connected. However there is no connected open subset that contains the point (0,1) and is contained within the open disk of radius 1 about this point,  and therefore the space is not locally connected, locally path-connected, locally simply-connected or locally contractible.

Proposition 1.14 Let X be a connected, locally path-connected topological space. Then X is path-connected.

Proof Choose a point x₀ of X. Let Z be the subset of X consisting of all points x of X with the property that x can be joined to x₀ by a path. We show that the subset Z is both open and closed in X.

Now, given any point x of X there exists a path connected open set N_x in X such that x ∈ N_x. We claim that if x ∈ Z then N_x ⊂ Z, and if x ∉Z then N_x ∩ Z = ∅

Suppose that x ∈ Z. Then, given any point x₀ of N_x, there exists a path in N_x from x₀ to x. Moreover it follows from the definition of the set Z that there exists a path in X from x to x₀. These two paths can be concatenated to yield a path in X from x₀ to x₀, and therefore x₀ ∈ Z. This shows that N_x ⊂ Z whenever x ∈ Z.  Next suppose that x ∉Z. Let x 0 ∈ N_x. If it were the case that x₀ ∈ Z, then we would be able to concatenate a path in N_x from x to x₀ with a path in X from x₀ to x₀ in order to obtain a path in X from x to x₀. But this is impossible, as x ∉Z. Therefore N_x ∩ Z = ∅ whenever x ∉Z.

Now the set Z is the union of the open sets N_x as x ranges over all points of Z. It follows that Z is itself an open set. Similarly X Z is the union of the open sets N_x as x ranges over all points of X Z, and therefore X Z is itself an open set. It follows that Z is a subset of X that is both open and closed. Moreover x₀ ∈ Z, and therefore Z is non-empty. But the only subsets of X that are both open and closed are ∅ and X itself, since X is connected.

Therefore Z = X, and thus every point of X can be joined to the point x₀ by a path in X. We conclude that X is path-connected, as required.

Let P be some property that topological spaces may or may not possess.

Suppose that the following conditions are satisfied:—

(i) if a topological space has property P then every topological space homeomorphic to the given space has property P;

(ii) if a topological space has property P then open subset of the given space has property P;

(iii) if a topological space has a covering by open sets, where each of these open sets has property P, then the topological space itself has property P.

Examples of properties satisfying these conditions are the property of being locally connected, the property of being locally path-connected, the property of being locally simply-connected, the property of being locally contractible,  and the property of being locally Euclidean.

Properties of topological spaces satisfying conditions (i), (ii) and (iii)  above are topological properties that describe the local character of the topological space. Such a property is satisfied by the whole topological space if and only if it is satisfied around every point of that topological space.

Definition Let f: X → Y be a continuous map between topological spaces X and Y . The map f is said to be a local homeomorphism if, given any point x of X, there exists some open set U in X with x ∈ U such that the function f maps U homeomorphically onto an open set f(U) in Y .

Lemma 1.15 Every covering map is a local homeomorphism.

Proof Let p: X˜ → X be a covering map, and let w be a point of X˜. Then p(w) ∈ U for some evenly-covered open set U in X. Then p⁻¹(U) is a disjoint union of open sets, where each of these open sets is mapped homeomorphically onto U. One of these open sets contains the point w: let that open set be U˜. Then p maps U˜ homeomorphically onto the open set U. Thus the covering map is a local homeomorphism.

Example Not all local homeomorphisms are covering maps. Let S¹ denote the unit circle in R², and let α: (−2, 2) → S¹ denote the continuous map that sends t ∈ (−2, 2) to (cos 2πt,sin 2πt). Then the map α is a local homeomorphism. But it is not a covering map, since the point (1, 0) does not belong to any evenly covered open set in S¹.

Let f: X → Y be a local homeomorphism between topological spaces X and Y , and let P be some property of topological spaces that satisfies conditions (i), (ii) and (iii) above. We claim that X has property P if and only if f(X) has property P. Now there exists an open cover U of X by open sets, where the local homeomorphism f maps each open set U belonging to U homeomorphically onto an open set f(U) in Y . Then the collection of open sets of the form f(U), as U ranges over U constitutes an open cover of f(X).  Now if X has property P, then each open set U in the open cover U of X has property P (by condition (ii)). But then set f(U) has property P for each open set U belonging to U (by condition (i)). But then f(X) itself must have property P (by condition (iii)). Conversely if f(X) has property P,  then f(U) has property P for each open set U in the open cover U of X. But then each open set U belonging to the open cover U has property P (as U is homeomorphic to f(U)). But then X has property P (by condition (iii)).  A covering map p: X˜ → X between topological spaces is a surjective local homeomorphism. Let P be some property of topological spaces satisfying conditions (i), (ii) and (iii) above. Then the covering space X˜ has property P if and only if the base space X has property P. A number of instances of this principle are collected together in the following proposition.

Proposition 1.16 Let p: X˜ → X be a covering map. Then the following are true:

(i) X˜ is locally connected if and only if X is locally connected;

(ii) X˜ is locally path-connected if and only if X is locally path-connected;

(iii) X˜ is locally simply-connected if and only if X is locally simply connected;

(iv) X˜ is locally contractible if and only if X is locally contractible;

(v) X˜ is locally Euclidean if and only if X is locally Euclidean.

Corollary 1.17 Let X be a locally path-connected topological space, and let p: X˜ → X be a covering map over X. Suppose that the covering space X˜ is connected. Then X˜ is path-connected.

Proof The covering space X˜ is locally path-connected, by Proposition 1.16.