Category
page 1Topological spaces
metric space
set equipped with a metric (distance function)
topological space
set of points and set of neighborhoods that satisfy axioms relating those points to those neighborhoods
number line
picture of a graduated straight line that serves as abstraction for real numbers; coordinate system in one-dimensional space
Klein bottle
type of non-orientable surface
Sierpiński triangle
fractal composed of triangles
Cantor set
fractal and set of points on a line segment
Menger sponge
3d fractal
Sierpinski carpet
plane fractal built from squares
topological vector space
vector space equipped with a compatible topology
discrete space
topological space whose every subset is both open and closed
simplicial complex
a geometrical object (set composed of points, line segments, triangles, and their n-dimensional counterpart) useful to describe certain topological spaces
trivial topology
topology where the only open sets are the empty set and the entire space
metrizable space
topological space that is homeomorphic to a metric space
pointwise convergence
notion of convergence in mathematics
CW complex
type of topological space
Sierpiński space
finite topological space with two points, only one of which is closed
Smith–Volterra–Cantor set
set that is nowhere dense (in particular it contains no intervals), yet has positive measure
hyperbolic space
homogeneous space that has a constant negative curvature (not any hyperbolic manifold)
topologist's sine curve
connected but not path-connected space; the graph of the function sin(1/𝑥) on the half-open interval (0,1] together with the origin
order topology
certain topology on totally ordered sets
lower limit topology
topology on the real numbers
Hilbert cube
infinite-dimensional cube with a compact topology
finite topological space
topological space with a finite number of points
pointed space
topological space with a distinguished point
loop space
space of basepoint preserving maps from a circle
Hawaiian earring
one-point compactification of a countably infinite family of open intervals
Sorgenfrey plane
product of two Sorgenfrey lines
long line
locally Euclidean topological space that is not paracompact
first uncountable ordinal
smallest ordinal number that, considered as a set, is uncountable
Cantor space
topological space
Moore plane
topological space
rose
topological space obtained by gluing together a collection of circles along a single point
Baire space
the Cartesian product of countably infinitely many copies of the set of natural numbers with the product topology
Antoine's necklace
topological embedding of the Cantor set in 3-dimensional Euclidean space
adjunction space
a common construction in topology where one topological space is attached or "glued" onto another
homology sphere
topological manifold whose homology coincides with that of a sphere, i.e. trivial except in the top and bottom degrees, where it has a single generator
box topology
mathematical concept
cocountable topology
Topology made of cocountable subsets
Knaster–Kuratowski fan
specific connected topological space
Tychonoff plank
topological space that is used in mathematics as a counterexample to several plausible-sounding conjectures
fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
comb space
pathological topological space
projectively extended real line
extension of the set of the real numbers by a point denoted ∞
pseudocircle
The pseudocircle is the finite topological space X consisting of four distinct points {a,b,c,d} with the following non-Hausdorff topology:
\{\{a,b,c,d\}, \{a,b,c\}, \{a,b,d\}, \{a,b\}, \{a\}, \{b\}, \varnothing\}.
compact convergence
type of mathematical convergence in topology