Glossary of Topology
Topology is a branch of mathematics concerned with those properties of
geometric configurations (as sets of points) which are unaltered by elastic
deformations (as a stretching or a twisting) that are homeomorphisms.
Also, a
topology is the set of all open subsets of a topological space.
Glossary
 Absolute neighborhood retract (ANR)
 Absolute retract
 a space A is an absolute retract if whenever a normal space X
has a closed subspace B homeomorphic to A, then B is a
retract of X. The fact that I and R¹ are absolute
retracts follows from the Tietze Extension Theorem.
 Accumulation point
 given a subset A of a topological space X, the point p
in X is called an accumulation point of A if each neighborhood
of p contains infinitely many distinct points of A.
 Adjoint chain mapping
 Alexandroff compactification
 Algebra
 an Ralgebra is an Rmodule with internal multiplication.
 Annulus
 Antipodal point
 Arc
 homeomorphic image of a closed line interval
 Automorphism
 an isomorphism of a group with itself
 Axiom of choice
 Baire category theorem
 Ball
 an rball in a metric space X is the open set N_{r}(x_{0})
= { x in X :  xx_{0}<r }.
 Banach space
 a complete normed vector space with either the real or complex numbers as
scalars
 Barycentric subdivision
 Base point
 a fixed point in a space for which maps of fixed point spaces map fixed
points to fixed points.
 Basic neighborhood space
 Basis
 a basis for a topological space X is a collection of open sets of X
that contains "arbitrarily small" neighborhoods of every point of X.
Specifically, for every point x of X, and for every open set U
containing x, the collection must include a neighborhood of x
lying within U.
 Basis for neighborhood systems
 Basis for open sets
 Bd
 a symbol used to denote the boundary operator, also seen is Bdry
prefix or ß operator
 Betti number
 Bidegree
 Bijection
 Blanket
 a simply connected cover of a sheet
 Bockstein homomorphism
 BolzanoWeierstrass theorem
 BorsukUlam theorem
 Boundary
 boundary of A is the union of points for which every open set
around them intersects both A and its complement OR the image of a
differential map in a complex.
 Boundary of Manifold
 the boundary Bd M of a manifold M is the set of points of M
that have boundary patches. The boundary of an nmanifold M is
an (n1) manifold ðM whose boundary ð(ðM) is empty {ø}.
 Boundary Operator
 a symbol ð used to denote the boundary of a chain complex see p226 in
Hocking
 Boundary patch
 a patch h:U−>H^{n} abount a point x
in U such that h(x) is in R^{n1},
(R^{n1} is itself a subset of H^{n})
 Bounded
 Bounded function
 Bounded subset
 Brouwer fixedpoint theorem
 if f is a continuous map of an n1 dimensional ball into
itself, then f has a fixed point or, every automorphism has a fixed
point.
 Cantor set
 Category
 Category (first or second)
 Cauchy Sequence
 Cellcomplex
 Centroid
 the center of mass of a figure. The centroid of a triangle is the
intersection of the medians.
 Chain
 any path in a manifold is a chain; it contains zero or more links. A chain
for a path ß is a chain that is path homotopic to ß.
 complex
 equivalent complex
 group
 homotopy
 map
 Character group
 Cl
 a symbol used to denote the closure operator, can also be indicated
by an overbar ¯ or macron accent or as superscript lowercase a ex: B^{a}.
 Clan
 Class
 Closure
 the closure of a subset A of a space X, denoted Cl A,
is the minimal closed set of X that contains A.
 Coboundary operator
 Cochain
 Cofibration
 Cofinal family of coverings
 Cohomology
 Cohomology groups
 Collar
 a pcollar is the union of all pneighborhoods of points on
a path, used in constructing a tube out of a path (as in knot theory).
 Compact
 a topological space is compact if every collection of open sets that
covers the space has a finite subset that also covers the space. The compact
subspaces of R^{n} are the closed and bounded sets.
 Countably
 Locally
 Para
 Pseudo
 Sequentially
 Space
 Subset
 Locally compact
 Compactum
 a metric space (X,d(,))is called a compactum if its
associated topological space is compact
 Complement
 Complete
 Complete metric
 Completely normal
 Completely regular
 Completely separable
 Complete system of neighborhoods
 Completion
 Component
 also calles 'connected component'. The components of a topological space
are its maximal connected subspaces. Two points of a space X lie in
the same component of X iff some connected subspace of X
contains both points
 Component space
 Component of a Point
 Composant
 Condensation point
 Cone complex
 Connected
 a topological space is connected if it cannot be partitioned into two
disjoint, nonempty open sets
 Arcwise
 Irreducibly
 Locally
 Multiply
 Path
 Polygonaly
 Subsets
 Simply
 Continuous
 a function where opens sets in the range "pull back" to open
sets in the domain
 Continuous transformation
 Contractible
 a space is contractible if it can be shrunk to a point within itself. The
homotopy that does this is called a 'contraction'. Contractible spaces are simply
connected
 Contractive
 Convex
 a subset X of R^{n} is convex if for every pair of
points in X, the line segment between them also lies in X.
 Convergence
 Coordinate
 Cobordism
 Countable
 Cover
 Open
 Sub
 Covering
 a collection of sets whose union is the whole space
 Open covering
 Covering map
 Covering space
 the domain of a covering map; also called a 'cover'. A space that
looks locally like the space it covers, but whose parts may be connected
together differently
 Covering transformation
 also called 'deck transformation'. A covering transformation is a
homeomorphism of a covering space with itself that preserves the covering
map. For any two liftings of a connected object, there is a covering
transformation that carries one to the other, provided that the covering
space is connected and locally pathconnected.
 Compactification
 Convolution
 a convolution of two planar regions is the set of all vector sums of a
point in one region with a point in the other
 Critical point
 Crosscap
 Cupproduct
 Cut
 Cut point
 Cutting set
 CWcomplex
 Cycle group
 Cylinder
 Dehn twist
 on a surface of genus g > 0, cut apart one of the handles along
a circle, give one handle a 360° twist, and glue the handles back together.
 Dedekind cut axiom
 Deformation retract
 a subspace A of a space X is a deformation retract if X
can be shrunk down to A without moving any point of A. The
homotopy that does the shrinking is called a 'deformation retraction'
 Degree of mapping
 Dense
 Denumerable
 deRham cohomology
 The nth deRham cohomology is the the space of nforms w with dw = 0,
modulo those of the form du where u is an (n1)form. Its dimension is
called the nth Betti number of the space.
 Derived set
 Diagonal set
 Diameter of a set
 Diffeomorphism
 a homeomorphism between manifolds which is also differentiable.
 Dimension
 Directed set
 Disc
 Disconnected
 Disconnecting subset
 Discrete topology
 a topological space X is discrete if every point of X is
open in X i.e. the integers form a discrete subspace of the real
line.
 Distance between sets
 Euclidean nspace
 or E^{n}, a metric space with distance function d(p,q)=((y_{1}x_{1})²+...+(y_{n}x_{n})²)^{½}
where p=(y_{1},...,y_{n}) and q=(x_{1},...,x_{n})
 Euler characteristic
 Euler's theorem
 EillenbergSteenrod axioms
 Embedding
 a mapping into a space whose image is homeomorphic to the domain. The
parametrization of a submanifold by means of a standard model. A knotted
sphere in 4space is an embedding of the familiar round sphere. Whitney's
theorem says that an dimensional manifold is guaranteed to have an
embedding in Euclidean space.
 Epimorphism
 Equivariant cobordism
 Exact
 a sequence is exact if the image of the map coming into an object is the
kernel of the outgoing map.
 Ext
 a symbol used to denote the Exterior operator
 Exterior
 Extension
 Extrinsic dimension
 the number of dimensions of the universe containing a subset  in other
words, a creature living outside the subset would "see" this
number of dimensions. The surface of a sphere has intrinsic dimension
2 and extrinsic dimension 3.
 Face
 the faces of an embedded planar graph are the regions into which the edges
of that graph divide the plane. The 'outer' face is the unique unbounded
ones.
 Fiber
 the inverse image of a point in the range
 Fiber bundle

 Filter
 Finite covering
 Finite intersection property
 First axiom of countability
 First countable
 First separation axiom
 Fixed point
 a point that is mapped to itself
 Flat
 a nmanifold is flat if it comes with a local embedding into R^{n}.
Flat Manifolds include sheets, blankets, and scraps of blankets
 Foliation
 Fr
 a symbol used to denote the frontier operator, also seen as a
germanic/stylized F prefix
 Fréchet's axiom (see First separation axiom)
 Freudenthal suspension
 Frontier
 the frontier of a subset A in a space X, denoted Fr A,
is Cl A  Int A: then set of points that lie in the closure of A
but not in the interior of A.
 Function
 a mapping from a domain to a range such that an element has only one image
 Functor
 a correspondence from one category to another mapping objects to objects
and preserving morphisms.
 Fundamental group
 the group of homotopy classes of loops at a base point of a space is a
topological invariant. It measures the holes in a space.
 GelfandFuks cohomology
 Graded module
 Group
 Group generator
 HahnMazurkiewicz theorem
 Haken manifold
 Hamiltonean path
 Hausdorff separation axiom
 for any two elements p and q of a topological space X
there exists disjoint open subsets P and Q in X such
that p is in P and q is in Q
 HausdorffBesicovitch dimension
 D_{H}(S) is the value of d for which the
Hausdorff measure of ddimensional volume of a set S namely h^{d}(S)
changes from infinity to zero, it is a space filling characterisation.
 Hausdorff measure
 Hausdorff space
 a Hausdorff space is defined by the property that every two distinct
points have disjoint neighborhoods
 HeineBorel theorem
 Hereditary property of a space
 Hilbert cube
 Homeomorphism
 a bijective continuous function of spaces with a continuous inverse. More
rigorously, two spaces X and Y are homeomorphic if there
exists a continuous bijection f:X−>Y such that the
inverse map f^{ 1}:Y−>X is also continuous
and preserves open and closed sets. To a topologist, homeomorphic spaces can
be transformed one into the other by an elastic deformation, so they are
"the same".
 Homology
 Homology groups
 Homomorphism
 A function that preserve the operators associated with the specified
structure.
 Homotopic paths
 Homotopic functions
 Homotopy
 a "continuous deformation" of one map to another or 'continuous
family' of maps, i.e. two maps f and g:X −> Y
are homotopic if there exists a continuous map F:X×I
−> Y such that F(x,0)=f(x) and F(x,1)=g(x).
 Homotopy class
 Homotopy groups, higher
 Homotopy type
 Hopfspace
 Ideal point
 Imbedding (???)
 Immersion
 A locally (but not globally) smoothly invertible mapping of one manifold
into another. The image may have selfintersections; the figure8 is an
immersion of the circle in 2D.
 Incidence number
 Indecomposable continuum
 Indentification
 Index of a transformation
 Injection
 a function which is onetoone, i.e. if f(x)=f(y),
then x=y.
 Int
 a symbol used to denote the interior operator, interior of set A
can also be denoted by A° or A^{o}
 Interior
 interior of A is the union of points which have an open set
containing them which is completely contained inside A. Also the
maximal open set contained in A.
 Intermediate value theorem
 Interval
 Intrinsic dimension
 the number of dimensions of the open subsets of a set  in other words, a
creature living inside a set would "see" this number of
dimensions. The surface of a sphere has intrinsic dimension 2 and
extrinsic dimension 3.
 Irreducible continuum
 Isometry
 a mapping of metric spaces which preserves the metric.
 Isomorphic
 Isomorphism
 Isotopy
 A homotopy of an object produced by a deformation of the ambient space, so
therefore the object cannot develop new selfintersections. The deformation
of the teapot to a torus is an isotopy, but the deformation to a point is
not.
 JordanBrouwer theorem
 every (n1)dimensional topological sphere divides E^{n}
into two parts
 Jordan curve
 Ktheory
 Klein bottle
 Knot
 A knot is defined as a closed, nonself intersecting line embedded in 3D.
 Kolmogorov's axiom
 Kuratowski closure
 Lebesque number
 Lift
 also called 'lifting'. With respect to a covering map p:M−>X,
a lift of a map a:C−>X is any map ã:C−>M
such that p º ã = a. The covering map p is the
covering of a sheet by its blanket.
 Lifting
 the process of converting maps into a base space to maps into its covering
space
 Limit point
 Link
 a path in a manifold that touches the manifold's boundary at its endpoints
alone.
 Link class
 an equivalence class under link homotopy; the set of links that are
linkhomotopic to a given link
 Link homotopy
 a homotopy between links that moves their endpoints along their respective
fringes; or the relation of being linkhomotopic. Two links are
linkhomotopic if there is a link homotopy (in the first sense) between them
 Lindenlöf's theorem
 Local
 a property of topological spaces is usually said to hold locally in
a space X if it holds within arbitrarily small neighborhoods of every
point of X. (For properties that open sets do not normally have, such
as compactness, the definition has to be modified somewhat.) For example, a
space is locally pathconnected if it has a basis of pathconnected
sets.
 Local embedding
 the map f:X−>Y is a local embedding if X
has a basis of open sets U such that f _{U} is
an embedding
 Local homeomorphism
 the map f:X−>Y is a local homeomorphism if X
has a basis of open sets U such that f(U) is open in Y
and f _{U} is an embedding
 Loop
 a map of a circle into a space OR a map of the unit interval [0,1] into a
space with the same beginning and end points, i.e. f(0)=f(1).
 Loop space
 Manifold
 a topological space which looks locally homeomorphic to R^{m}
for some m. m is called the intrinsic dimension of the
manifold. It is a A generalization of ndimensional space in
which a neighborhood of each point, called its chart, looks like Euclidean
space. The charts are related to each other by Cartesian coordinate
transformations and comprise an atlas for the manifold. The atlas may be
nontrivially connected; there are roundtrip tours of a manifold that
cannot be contracted to a point.
 Mapping cone
 Mapping cylinder
 MayerVietoris sequence
 Metacompactness
 Metric
 also called 'distance metric', a metric on a set P is a funtion d:P×P−>R^{+}
satisfying three axioms:
 d(p,q)=0 <=> p=q
(the distance between two points is zero if and only if the two points are
the same)
 d(p,q)=d(q,p) for all p,q
in P (the distance between p and q is
the same and the distance between q and p)
 d(p,q)+d(q,r) >= d(p,r)
for all p,q,r in P (the sum of the distances
between p and q and between q and r is greater
than or equal to the distance between p and r)
The metric d gives rise to a topology on P; a basis for this
topology is the collection of sets { q: d(p,q)<e}
for p in P and e>0. In other words, a subset S of P
is open in the metric topology if for every point p in S there
is a number e>0 such that S contains the set { q: d(p,q)<e}.
 Metric space (see Metric above)
 a space with a "distance" function, called a metric, defined
above.
 Metrically equivalent
 Minimal Surface
 A surface that locally has the smallest area given a particular
topological shape for it, and possibly, constrained by a fixed boundary
(soapfilms) or prescribed behavior at infinity.
 Monomorphism
 Multiply connected
 not simply connected
 Complete metric space
 Mesh (of a complex)
 Metrizable
 Möbius strip
 morphism
 automorphism
 bijective endomorphism
 diffeomorphism

 endomorphism
 G−>G
 epimorphism
 onto or surjective
 homeomorphism

 homomorphism

 isomorphism
 1 to 1 or bijective
 monomorphism
 injective
 Multiply connected
 Mutually separated
 Neighborhood
 a neighborhood of a point or a set is an open set that containes it.
 Neighborhood space
 Nerve of a covering
 Net
 Nonorientable
 Noncut point
 Normal
 said of topological spaces. In a normal space, every two disjoint closed
sets have disjoint neighborhoods. All metric spaces are normal.
 Normal space
 Nowhere dense subset
 nsphere
 Obstruction ?
 Open mapping
 Onepoint compactification
 Orientable
 Oriented
 Paracompactness
 Partition
 Patch
 a patch about a point x in an nmanifold is a homeomorphism
of a neighborhood of x with an open set in the halfspace H^{n}
 Path
 a continuous function with domain I=[0,1]
 class
 an equivalence class under the relation of path homotopy
 closed
 component
 the path components of a topological space are its maximal path
connected subsets. Two points lie in the same path component of a space X
iff there is a path in X from one point to the other.
 constant
 initial point of a
 product of
 terminal point of a
 Pathconnected
 a topological space is pathconnected if every pair of its points can be
connected by a path. Every pathconnected space is connected, but not vice
versa. If a space is connected and locally pathconnected, however,
then it is path connected.
 Path homotopy
 a homotopy between paths that fixes their endpoints; or the relation of
being pathhomotopic. Two paths are pathhomotopic if there is a path
homotopy between them.
 Planar network
 Point
 accumulation
 antipodal
 base
 condensation
 critical
 cut
 end
 fixed
 ideal
 image
 initial
 limit
 noncut
 set
 separable
 separate
 terminal
 Pointwise convergent
 Product set
 Product space
 Projection
 usually refers to composition with the covering map. For example, if ß
is a path in a blanket M, and p:M−>S is the
covering map, then the projection of ß is the path pºß:I−>S
 Projective plane
 Pseudocompactness
 Pseudometric
 Quasicomponent
 Quotient space
 a space obtained from another by identifying or "gluing" some
points to some others. Formally, Y is a quotient space of X if
there is a surjective map f:X−>Y such that the open
sets of Y are those subsets U of Y for which f^{
1}(U) is open in X
 Quotient group
 Rational density theorem
 Reduced suspension
 Refinement
 Regular
 Relation type
1. 
reflexive 
xRx 
2. 
irreflexive 
~xRx or not 1 
3. 
symmetric 
xRy<=>yRx 
4. 
antisymmetric 
(xRy and yRx)=>x=y 
5. 
transitive 
(xRy and yRz)=>xRz 
6. 
connected 
x<>y=>(xRy or yRx) 
7. 
left unique 
(xRz and yRz)=>x=y (a.k.a
functional) 
8. 
right unique 
(xRy and xRz)=>y=z 
9. 
biunique 
leftright unique: 7 and 8 
10. 
equivalence 
1, 3, 5 and support is S 
11. 
partial order 
1, 4 and 5 
12. 
total order 
1, 4, 5 and 6 (a.k.a. linear order) 
 Relatively closed
 Relatively open
 Relative neighborhood
 Relative topology
 Residual
 Restriction
 Retract
 a subspace A of a space X is a retract if there is a map f:X−>A
that fixes every point of A. The map f is called a
'retraction'.
 Retraction Mapping
 Rimcompact space
 Ring
 Deformation retract
 Scrap
 a simply connected, open manifold of a blanket
 Second axiom of countability
 Second Countabile
 Seifert fibered
 Separable
 a separable space is one that has a countable dense subset, that is a
countable subset whose closure is the whole space.
 Separate points
 Separation axioms
 Set
 compact
 complete
 connected
 bounded
 closed
 empty ø
 open
 null
 Sheet
 Sierpinski Space
 Simplectic structure
 Simplicial complex
 Simply connected
 a topological space is simply connected if
 it is path connected
 every loop in that space can be continuously shrunk to a point.
 Singleton
 Singularity
 Smoothing
 Space
 a topological space: a set with a system of neigborhoods (open sets)
closed under finite intersection and arbitrary union.
 Hausdorff space
 Lashnev space
 Suslin space
 Luzin space
 FréchetUrison space
 Lindenlöf space
 Spectra
 Split
 Starlike space
 Steepest Descent Method
 A particular way of guiding an isotopy of an embedded surface to one which
minimizes a function that measures its shape. Moving down the gradient of
the area function often terminates at a minimal surface.
 StoneCech compactification
 Straight
 a path in a flat mmanifold is straight if itd projection to R^{m}
is linear and nonconstant
 Subbasis
 Subcovering
 Subcontinuum
 Subgroup
 Subnet
 Submanifold
 a subset of a manifold that is itself a manifold
 Subpath
 a subpath of a path µ is any path of the form µ_{s}_{:t}
for s,t in I. Defined: µ_{s}_{:t}(x)=µ((1x)s+xt).
 Subsequence
 Subspace
 a subset A of a topological space X with the inherited
topology: the open set in A are the intersections of the open sets of
X with A
 Support
 a straight path ß in R² supports a piecewise linear path w
at s in (0,1) if ß(1)=w(s) and w turns
towards ß(0) at s. If w_{r:s} and w_{s:t}
are segments of w, we also say that ß supports these segments
 Surface symbol
 Surgery
 Surjection
 a function where every element in the range has a preimage in the domain.
 Suspension
 T_{0}  T_{4}.
 Tame
 Tangent bundle
 Tangent space
 Taxicab norm
 defined on R² by d(x, y) = x + y
 TikhonovUryson theorem
 Tietze's extension theorem
 Top
 category of all topological groups?
 Topology
 a collection of "open sets" closed under unions and finite
intersections.
 algebraic
 differential
 point Set
 coarser
 discrete
 finer
 identification
 metric
 order
 product
 subspace
 trivial
 Topologically equivalent
 Topological property
 a property that is preserved by homeomorphisms; what topology is all
about.
 Topological space
 Topological space associated with a metric space
 Topological time series analysis
 Tor
 Torsion
 an Rmodule has torsion if some nonzero element, a, and
some nonzero scalar, r, imply ra = 0.
 Torus
 the surface of a donut, i.e. a rectangle with the top and bottom
identified to form a cylinder and the the ends of the cylinder identified to
form a "donut". Constructed as the cartesian product × of two
1spheres (circles) T ² = S ¹×S ¹
 tree
 a tree is a graph with the property that there is a unique path from any
vertex to any other vertex traveling along the edges.
 Tychonoff space
 Tychonoff theorem
 Torus
 Ultrafilter
 Uncountable
 Uniformly continuous
 Uniform convergence
 a sequence of functions <f_{n}> into a metric
space with metric d converges uniformly to a function f if
for every e>0 there is an N such that d(f(x),
f_{n}(x)<e for all n>N
and all x.
 Uniform space
 Unit ncube (a.k.a. I^{n})
 Universal covering space
 Urysohn's metrization theorem
 Vector field
 Vietoris homology group
 Weaker
 Whitney Trick
 In five (or higher) dimensions, there's a pretty neat move called the
"Whitney Trick" that allows you to move complicated objects past
each other and separate them out into understandable pieces.
 Zorn's lemma
Related pages in this website
Geometry Glossary
Number Theory Glossary
Statistics Glossary
The webmaster and author of this Math Help site is
Graeme McRae.