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 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
- Graded module
- 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
- 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
link-homotopic to a given link
- Link homotopy
- 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:
- 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
(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
- 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
- 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
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