Euclidean Space - Topology of Euclidean Space

Topology of Euclidean Space

Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric. The metric topology on En is called the Euclidean topology. A set is open in the Euclidean topology if and only if it contains an open ball around each of its points. The Euclidean topology turns out to be equivalent to the product topology on Rn considered as a product of n copies of the real line R (with its standard topology).

An important result on the topology of Rn, that is far from superficial, is Brouwer's invariance of domain. Any subset of Rn (with its subspace topology) that is homeomorphic to another open subset of Rn is itself open. An immediate consequence of this is that Rm is not homeomorphic to Rn if mn — an intuitively "obvious" result which is nonetheless difficult to prove.

Read more about this topic:  Euclidean Space

Famous quotes containing the word space:

    I would have broke mine eye-strings, cracked them, but
    To look upon him, till the diminution
    Of space had pointed him sharp as my needle;
    Nay, followed him till he had melted from
    The smallness of a gnat to air, and then
    Have turned mine eye and wept.
    William Shakespeare (1564–1616)