nice topological space
nice category of spaces
convenient category of topological spaces
Freudenthal suspension theorem
CW-complex, Hausdorff space, second-countable space, sober space
compact space, paracompact space
connected space, locally connected space, contractible space, locally contractible space
topological vector space, Banach space, Hilbert space
point, real line, plane
sphere, ball, annulus
loop space, path space
Cantor space, Sierpinski space
long line, Warsaw circle
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For n∈ℕ a natural number, write ℝ n for the Cartesian space of dimension n. The Euclidean topology is the topology on ℝ n characterized by the following equivalent statements
it is the metric topology induced from the canonical structure of a metric space on ℝ n with distance function given by d(x,y)=∑ i=1 n(x i−y i) 2;
an open subset is precisely a subset such that contains an open ball around each of its points;
it is the product topology induced from the standard topology on the real line.
Two Cartesian spaces ℝ k and ℝ l (with the Euclidean topology) are homeomorphic precisely if k=l.
A proof of this statement was an early success of algebraic topology.