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### 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.
Alexandroff compactification
Algebra
an R-algebra is an R-module 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 r-ball in a metric space X is the open set Nr(x0) = { x in X : | x-x0|<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
Bolzano-Weierstrass theorem
Borsuk-Ulam 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 n-manifold M is an (n-1) 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−>Hn abount a point x in U such that h(x) is in Rn-1, (Rn-1 is itself a subset of Hn)
Bounded
Bounded function
Bounded subset
Brouwer fixed-point theorem
if f is a continuous map of an n-1 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
Cell-complex
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: Ba.
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 p-collar is the union of all p-neighborhoods 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 Rn 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 Rn 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 path-connected.
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
Cup-product
Cut
Cut point
Cutting set
CW-complex
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 n-forms w with dw = 0, modulo those of the form du where u is an (n-1)-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 n-space
or En, a metric space with distance function d(p,q)=((y1-x1)²+...+(yn-xn)²)½ where p=(y1,...,yn) and q=(x1,...,xn)
Euler characteristic
Euler's theorem
Eillenberg-Steenrod 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 4-space 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 n-manifold is flat if it comes with a local embedding into Rn. 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.
Gelfand-Fuks cohomology
Group
Group generator
Hahn-Mazurkiewicz 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
Hausdorff-Besicovitch dimension
DH(S) is the value of d for which the Hausdorff measure of d-dimensional volume of a set S namely hd(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
Heine-Borel 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
Hopf-space
Ideal point
Imbedding (???)
Immersion
A locally (but not globally) smoothly invertible mapping of one manifold into another. The image may have self-intersections; the figure-8 is an immersion of the circle in 2D.
Incidence number
Indecomposable continuum
Indentification
Index of a transformation
Injection
a function which is one-to-one, 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 Ao
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 self-intersections. The deformation of the teapot to a torus is an isotopy, but the deformation to a point is not.
Jordan-Brouwer theorem
every (n-1)-dimensional topological sphere divides En into two parts
Jordan curve
K-theory
Klein bottle
Knot
A knot is defined as a closed, non-self intersecting line embedded in 3-D.
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
a path in a manifold that touches the manifold's boundary at its endpoints alone.
a homotopy between links that moves their endpoints along their respective fringes; or the relation of being link-homotopic. Two links are link-homotopic 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 path-connected if it has a basis of path-connected 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 Rm for some m. m is called the intrinsic dimension of the manifold. It is a A generalization of n-dimensional 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 non-trivially connected; there are round-trip tours of a manifold that cannot be contracted to a point.
Mapping cone
Mapping cylinder
Mayer-Vietoris sequence
Metacompactness
Metric
also called 'distance metric', a metric on a set P is a funtion d:P×P−>R+ satisfying three axioms:
1. d(p,q)=0 <=> p=q (the distance between two points is zero if and only if the two points are the same)
2. 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)
3. 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 (soap-films) 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
Non-orientable
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
n-sphere
Obstruction ?
Open mapping
One-point compactification
Orientable
Oriented
Paracompactness
Partition
Patch
a patch about a point x in an n-manifold is a homeomorphism of a neighborhood of x with an open set in the half-space Hn
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
Path-connected
a topological space is path-connected if every pair of its points can be connected by a path. Every path-connected space is connected, but not vice versa. If a space is connected and locally path-connected, however, then it is path connected.
Path homotopy
a homotopy between paths that fixes their endpoints; or the relation of being path-homotopic. Two paths are path-homotopic 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
Quasi-component
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 left-right 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
Rim-compact 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
1. it is path connected
2. 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
Fréchet-Urison 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.
Stone-Cech compactification
Straight
a path in a flat m-manifold is straight if itd projection to Rm 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 µs:t for s,t in I. Defined: µs:t(x)=µ((1-x)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 wr:s and ws: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
T0 - T4.
Tame
Tangent bundle
Tangent space
Taxicab norm
defined on R² by d(x, y) = |x| + |y|
Tikhonov-Uryson 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 R-module has torsion if some non-zero element, a, and some non-zero 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 1-spheres (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 <fn> 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), fn(x)<e for all n>N and all x.
Uniform space
Unit n-cube (a.k.a. In)
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.