topological space
set of points and set of neighborhoods that satisfy axioms relating those points to those neighborhoods
first-countable space
topological space in which every point admits a countable neighborhood basis
sequentially compact space
Topological space where every sequence has a convergent subsequence.
connected space
topological space that cannot be written as the disjoint union of two nonempty open subsets
countably generated space
topological space in which the topology is determined by its countable subsets
quotient space of an algebraic stack
topological space associated to an algebraic stack F, which as a set is the set of all integral substacks of F and which is given a “Zariski topology”: an open subset has a form |U| ⊂ |F| for some open substack U of F
configuration space
moduli space of n points on a space M; if M is a manifold, in general forms an orbifold
CW complex
type of topological space
separable space
topological space with a dense countable subset
dense-in-itself space
topological space with no isolated point
suspension
quotient space; operation of suspension creates a way of moving up in dimension (dimension n + 1)
Schwartz space
function space of all functions whose derivatives are rapidly decreasing
connected sum
geometric modification on manifolds
rose
topological space obtained by gluing together a collection of circles along a single point
partially ordered space
topological space equipped with a closed partial order
ringed space
space together with a sheaf of rings
locally compact space
topological space such that every point has a neighbourhood with compact closure
totally disconnected space
topological space in which all connected components are singletons
H-space
topological space with a multiplication that is associative and unital up to homotopy
join
topology term
Moore space
topological space with only one nonzero reduced homology group
Infinite broom
Classic counterexample in topology
spectrum of a C*-algebra
the set of unitary equivalence classes of irreducible *-representations of A, equipped with a natural topology; for the C*-algebra of continuous real-valued functions on a compact Hausdorff space, this recovers the original topological space
homogeneous space
topological space that is the space of cosets of a topological group
cylinder
topological construction, a product with closed interval
spectrum of a ring
set of a ring's prime ideals
topological vector space
vector space equipped with a compatible topology
solenoid
compact connected topological space
topological monoid
Concept in mathematics
manifold
topological space that at each point resembles Euclidean space (unspecified type)
locally connected space
property of topological spaces
topological ring
ring where ring operations are continuous
mapping cylinder
topological construction
classifying space
topological space equipped with a principal bundle with the property that any principal bundle (with the same fiber group) over a paracompact manifold is isomorphic to a pullback of the principal bundle over this topological space
G-space
topological space equipped with a group action
mapping cone
topological construction
compactly generated space
topological space
loop space
space of maps from a circle
Dunce hat
compact topological space
topological semigroup
semigroup with continuous operation
universal space
topological space that contains a homeomorphic image of every topological space of a certain class
relatively compact subspace
subset of a topological space whose closure is compact
Cantor space
topological space
regular space
topological space in which a point and a closed set are, if disjoint, separable by neighborhoods
fixed-point space
topological space such that every endomorphism has a fixed point