ENTRY TOPOLOGY
Authors: Oliver Knill 2003, John Carlson 2003-2004
Literature: http://at.yorku.ca/cgi-bin/bell/props.cgi
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| Alexander compactification
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The Alexander compactification Y of a Hausdorff space (X,O) is the
topological space (Y = X cup x,P), where x is an additional point. The
topology P consists of the elements in O and the complements of
closed subsets as neighborhoods of that point.
The new topological space Y is compact.
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| Alexander's subbase theorem
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The Alexander's subbase theorem: if every open cover of a
topological space X has a finite sub-cover then X is compact.
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| arc-connected
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A topological space is called arc-connected
if any
two points can be connected by a path, a continuous
image of an interval. Path connected is
stronger than connected but not equivalent: the subset
(x,sin(1/x)), x in R^+ cup (0,y), -1 leq y leq 1
of the plane with topology induced from the plane is connected
but not path connected. Arc-connected is also called path-connected.
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| Baire category
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Baire category is a measure for the size of a set in
a topological space. Countable unions of nowhere dense sets
are called of the first categorie or meager, any other set of
second category. Complements of meager sets are called residual.
Baire category is used to quantify certain sets. For example it is
known that "most" numbers are Liouville numbers in the sense that
they form a residual set among all real numbers.
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| Baire space
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A Baire space is a topological space with the property
that the intersection of countable family of open dense subsets
is dense.
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| Baire category theorem
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The Baire category theorem: a complete metric space is
a Baire space.
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| ball
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A ball in a metric space is a set of the form
y | d(x,y) 1 is connected
but not locally connected because small neighborhoods of the point
(0,1) are not connected.
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| Hausdorff
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A topological space (X,T) is called Hausdorff if for every
two points x,y in X, there are disjoint open sets U,V in T such
that x in U and y in V. This is refined through
seperation axioms, T0, ..., T4. Hausdorff is also called T2.
Any metric space is Hausdorff: if d is the distance betwen x and y,
then balls of radius d/3 around x and y seperate the points.
The plane X with semimetric d(x,y) = |x_1-y_1| is not Hausdorff:
the points x=(0,-1) and y=(0,1) can not be seperated by open sets.
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| seperation axioms
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seperation axioms define classes of topological spaces with decreasing
seperability properties:
T4 Rightarrow T3 Rightarrow T2 Rightarrow T1 Rightarrow T0.
T0 space: for two different points x,y in X one of the points has
an open neighborhood U not containing the other point.
T1 space: for two different points x,y in X there exists an open
neighborhood U of x and an open neighborhood V of y.
such that x is not in V and y is not in U.
T2 space: also called Hausdorff" two different points x,y
can be seperated with disjoint neighborhoods U,V.
T3 space: T1 and regular: any point x and any closed set F not containing
x can be seperated by two disjoint neighborhood.
T4 space: T1 and normal: any two disjoint sets F,G can be separated
by two disjoint open sets.
It is known that a T4 space with a countable basis is metrizable.
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| Hausdorff topology
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The Hausdorff topology is a metric on the set of closed bounded
subsets of a complete metric space. The distance between two
sets A and B is the infimum over all r for which A is contained in
a r-neighborhood of B and B is contained in a r-neighborhood of A.
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| Lindeloef
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A topological space is called Lindeloef if every
open cover of X contains a countable subcover.
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| compact
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A topological space is called compact if every open
cover of X contains a finite subcover. Examples of results known
about compactnes:
Heine-Borel theorem: a closed interval in the real line is compact.
If f: X to Y is continuous and onto and X is compact, then
Y is compact. As a consequence, a continouous function on a compact
subspace has both a maximum and a minimum.
In a Hausdorff space, compact sets are closed.
In a metric space, compact sets are closed and bounded.
Closed subsets of compact spaces are compact.
Tychonof theorem: the product of a collection of compact spaces is compact.
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| countably compact
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A topological space is called countably compact if every
countable open cover of X contains a finite subcover.
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| locally compact
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A topological space is called locally compact if every point
has a neighborhood, which has a compact closure. Examples.
The real line is compact but not locally compact.
A compact Hausdorff space is locally compact.
The n-dimensional Euclidean space R^n is lcally compact but not compact.
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| locally compact
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A set U of open sets in a topological space (X,O)
is called locally finite if every point x in X has
a neighborhood V, such that V has a nonempty intersection
with only finitely many elements in U.
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| paracompact
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A topological space (X,O) is called paracompact
if every open cover has a countable, locally finite subcover.
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| relatively compact
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A subset A of a topological space (X,T) is called relatively compact
if the closure of A is compact.
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| filter
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A filter on a nonempty set X is a set of subsets F satisfying
X is in F, but the empty set emptyset is not in F.
If A and B are in F, then their intersection is in F.
If A is in F and B is a subset of A, then B is in F.
Examples:
Principal filter for a nonempty subset A consists of all
subsets of X which contain A.
Frechet filter for an infinite set consists of all
subsets of X such that their complement is finite.
Neighborhood filter of a point x in a topological space
(X,T) is the set of open neighborhoods of x.
Elementary filter for a sequence x_n in X consists of
all sets A in X such that x_n is in A for large enough n.
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| converges
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A sequence x_n in a topological space converges to a point x,
if for every neighborhood U of x, there exists an integer m, such that for
n>m one has x_n in U.
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| Filter convergence
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Filter convergence A filter F converges to x in a topological space
(X,T) if F contains the neighborhood filter G of x, that is if F contains
all neighborhoods of x. For example, an elementary filter to a sequence x_n
converges to a point x, if and only if x_n converges to x.
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| accumulation point
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A point y is called an accumulation point of a filter F, if there
exists a filter G containing F such that G converges to x.
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| directed
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A set M is called directed if there exists a partial order (M,<)
on M satisfying for every two points a,b in M there exists c, with
a0, B_r(x)= y | d(x,y) X,
where D is a directed set.
For example: if D is the set of natural numbers, then a net is a sequence.
A net defines a filter F: it is the set of all sets A such that x_t
is eventually in A. A net x_t converges to a point x if and only if the
associated filter converges to x.
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| open cover
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open cover A subset U of O, where (X,O) is a topological space
is called an open cover of X if the union of all elements in U
is X. If U and V are open covers and V subset U, then V
is called a subcover of U.
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| product space
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The product space between topological spaces is defined as
(X x Y, O x P), where X x Y is the set of all
pairs (x,y), x in X, y in Y and O x P is the coarsest
topological space which contains all products A x B, where A in O
and B in P. For example, if (X,O)=(Y,P) are both the real line with
the topology generated by d(x,y) = |x-y|, then the product space is
homeomorphic to the plane with the metric
d(x,y) = sqrt(x_1-x_2)^2+(y_1-y_2)^2.
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| second countable
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A topological space is called second countable,
if it has a countable basis.
Example. Every seperable metric space is second countable.
Especially, every finite-dimensional Euclidean space is
second countable.
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| metrizable
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A topological space is called metrizable if there exists
a metric d on the set X that induces the topology of X.
Any regular space with a countable basis is metrizable.
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| homotopic
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homotopic If f and g are continuous maps from the topological space
X to a topological space Y, we say that f is homotopic
to g if there is a continuous map F from X x I to Y,
such that F(x,0) = f(x) and F(x,1) = g(x) for all x.
For example, the maps f(x) = x^2 and g(x) = sin(x) on the
real line are homotopic, because we can define
F(x,t) = (1-t)x^2 + t sin(x). The maps
f(x) = x and g(x) = sin(2 pi x) on the circle are not
homotopic. While g is homotopic to the constant function h(x)=0,
the map f(x) can not be deformed to a constant without breaking
continuity.
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| induced topology
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The induced topology on a subset A of X, where (X,T) is a topological
spoace is the the topological space (A, Y cap A _Y in T).
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| path homotopic
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path homotopic If f and g are and continuous homotopic maps from an interval
to a space X, we say f and g are path homotopic if their
images have the same end points. For instance, the maps
f(x) = x^2 and g(x) = x^3 are path homotopic on the closed
interval from 0 to 1. The maps f(x)=2 x^2 and g(x)=x^3 are homotopic on
the unit interval but not path homotopic.
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| loop
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A loop is a path in a topological space that begins and ends
at the same point. A loop is also called a closed curve. Loops play
a role in definitions like simply connected: a topological space is
simply connected if every loop is homotopic to a constant loop which
is a fancy way telling that every closed path can be collapsed inside X
to a point.
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| fundamental group
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The fundamental group of a topological space at a point is
the set of homotopy classes of loops based at that point.
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| Topologist's Sine Curve
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The Topologist's Sine Curve is the union S of the graph of the function
sin(1/x) on the positive real axes R^+ with the y-axes.
It an example of a topological space which is connected but not path-connected.
Proof: if S were path-connected, there would exist a
path r(t)=(x(t),y(t)) connecting the two points (0,1) and (0,pi).
The set t | r(t) in S is closed. Let T be the largest t in that
set for which r(t) is in the y-axes. Then x(T)=0 and r(t)=(x(t),sin(1/x(t)) for t>T.
Because there are times t_n >t_n-1>T, t_n to T for which y(t_n)= (-1)^n,
the function r(t) can not be continuous at t=T.
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| Urysohn lemma
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The Urysohn lemma tells that if X is a normal space and
A and B are disjoint closed subsets of X, then there exists a
continuous map f from X to the unit interval such that f(x) = 0
for all x in A, and f(x) = 1 for all x in B.
Proof: use the normality of X to construct a family
U_p of open sets of X indexed by the rational numbers P
in the unit interval so that for p