Simply Connected Space - Formal Definition and Equivalent Formulations

Formal Definition and Equivalent Formulations

A topological space X is called simply connected if it is path-connected and any continuous map f : S1 → X (where S1 denotes the unit circle in Euclidean 2-space) can be contracted to a point in the following sense: there exists a continuous map F : D2 → X (where D2 denotes the unit disk in Euclidean 2-space) such that F restricted to S1 is f.

An equivalent formulation is this: X is simply connected if and only if it is path-connected, and whenever p : → X and q : → X are two paths (i.e.: continuous maps) with the same start and endpoint (p(0) = q(0) and p(1) = q(1)), then p and q are homotopic relative {0,1}. Intuitively, this means that p can be "continuously deformed" to get q while keeping the endpoints fixed. Hence the term simply connected: for any two given points in X, there is one and "essentially" only one path connecting them.

A third way to express the same: X is simply connected if and only if X is path-connected and the fundamental group of X at each of its points is trivial, i.e. consists only of the identity element.

Yet another formulation is often used in complex analysis: an open subset X of C is simply connected if and only if both X and its complement in the Riemann sphere are connected.

The set of complex numbers with imaginary part strictly greater than zero and less than one, furnishes a nice example of an unbounded, connected, open subset of the plane whose complement is not connected. It is nevertheless simply connected. It might also be worth pointing out that a relaxation of the requirement that X be connected, leads to an interesting exploration of open subsets of the plane with connected extended complement. For example, a (not necessarily connected) open set has connected extended complement exactly when each of its connected components are simply connected.

Read more about this topic:  Simply Connected Space

Famous quotes containing the words formal, definition and/or equivalent:

    The formal Washington dinner party has all the spontaneity of a Japanese imperial funeral.
    Simon Hoggart (b. 1946)

    Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.
    Elaine Heffner (20th century)

    Nobody can deny but religion is a comfort to the distressed, a cordial to the sick, and sometimes a restraint on the wicked; therefore whoever would argue or laugh it out of the world without giving some equivalent for it ought to be treated as a common enemy.
    Mary Wortley, Lady Montagu (1689–1762)