Topology
There are various approaches to topology--point-set, algebraic, geometric. This page assumes some general, basic understanding of mathematics. Really, as long as you aren't actually math-phobic, it'll be fine.
Y Getting the Preliminaries out of the Way
- If A and B are sets, then a subset R of the Cartesian product of those sets (R Í A ´ B) is a relation from A to B. So, a relation on A is simply a subset of A ´ A. The domain of R is Dom(R) = {x Î A : $ y Î B with (x, y)Î R}. Similarly, the range of R is Rng(R) = {y Î B : $ x Î A with (x, y) Î R}.
- An equivalence relation on A is a relation on A that is reflexive (" x Î A, (x, x)ÎR), symmetric (" x, y Î A, (x, y)ÎR), and transitive (" x, y, z Î A, (x, y), (y, z) Î R Þ (x, z) Î R). The equivalence class of x Î A is R_{x} = {y Î A : (x, y) Î R), often read "x mod R" or, for the fussy ones, "x modulo R". The set all all equivalence classes is R_{A} = {R_{x} : x Î A }, read "A modulo R".
- Recall that a directed graph, or digraph, tells us which way we're going on an edge of a graph. It'd be pretty easy to construct a digraph where, whenever (x, y)Î R, there's an edge from x to y, x® y. Such a graph would have all vertices looping back to themselves (reflexive). Unless all members of A are related to all other members of A, the graph would also be disconnected into group of elements (vertices) that are related to each other but, thanks to the transitive property, not related or bridged to the other members. Be careful, though. The disconnected subgraphs are cliques iff R is an equivalence relation because the symmetric and transitive properties combined will force each equivalence class to form a complete subgraph.
- If A is set, the a partition of A is a family of subsets of A, A = {S_{i} : S_{i} Í A}, such that (i) Æ Ï A , (ii) S_{i}, S_{j} Î A Þ either S_{i} = S_{j} Y or S_{iÇ }S_{j} = Æ , and (iii) È _{XÎ }A = A. These properties respectively ensure that the partition (i) contains non-empty subsets, (ii) contains disjoint subsets, and (iii) covers the entire set A. Now, a partition can define an equivalence relation, R, by simply "relating" elements of a subset to each other. That is, each subset (member of A ) is an equivalence class unto itself. So, (x, y)ÎR iff $ S_{i} Î A ' x, y Î S_{i}. Notice that the definition here sorta hints at some sort of enumeration--but that doesn't mean there's necessarily a methodological ordering to the enumeration of the subsets. It's just a bit nice to put them into something of a list sometimes. Anyway, it's good to see the connection between partitions and equivalence classes.
- A function (mapping), f, from A to B is a relation from A to B such that (i) Dom(f) = A and (ii) (x, y), (x, z )Î f Þ y = z. So, f Í A ´ B usually written f:A® B.
- Given a set A, the power set of A is P(A) = {S : S Í A}, the set of all subsets of A.
- Given a set E, a metric is a function to the Real numbers, d:E ´ E ® R, such that (i) " x, y Î E, d(x, y) ³ 0 and d(x, y) = 0 iff x = y, (ii) " x, y Î E, d(x, y) = d(y, x), and (iii) " x, y, z Î E, d(x, z) £ d(x, y) + d(y, z). The notion of a metric is...well, think of how SI (système internationale) is our most convenient, scientific, base-10 system of measurement, often referred to as "the" metric system. Really, a metric can be considered to be a distance function. The first condition (i) ensures that no distance is less than zero. The second condition (ii) is a somewhat symmetric--if your office is four miles from your house, then your house is four miles from your office. The final condition (iii) is known as the triangle inequality, and it helps add even more common sense to the notion of a distance function. Sometimes, we refer to a metric space by means of an ordered pair, (E, d).
(*) E = R, d(x, y) = |y - x| (the usual metric, a special case of the Euclidean metric below)
(*) E = R^{n}, d(x, y) = (the Euclidean metric) The notation of R^{n}, refers to the arrays, n-tuples, single column matrices of Euclidean n-space with elements or points often denoted like x = (x_{1}, x_{2},... x_{n}). Notice that, if n = 1, then this is just the usual metric.
(*) Given a set A, define E as the set of all Real-valued, bounded functions of A (E = {f : f :A® R, is bounded}) and d is defined by d(f, g) = sup_{xÎ A}|g(x) - f(x)| " f, g Î E. (the uniform metric)
(*) Given a set E, define a metric by . (the discrete metric)
- For a metric space (E, d), a Î E, and r Î R, the r-ball (neighborhood) with center a is V_{d} (a, r) = {x Î E : d(a, x) < r}. The deleted (or punctured) r-ball with center a is V_{d}^{'} (a, r) = {xÎ E : 0 < d(a, x) < r}.
Y Topological Spaces
Given a set E, a topology on E is T Í P(E) such that (i) E, Æ Î T, (ii) S_{1},...,S_{n} Î T Þ È _{i}=1,...n S_{i} Î T, and (ii) S_{1},...,S_{n} Î T Þ Ç _{i}=1,...n S_{i} Î T. [Note: The second condition does not require that the family of sets be finite, but the third condition does.} A topological space is an ordered pair, (E, T), where E is the "underlying set" and T is the topology on E. If T and T' are topologies on E and T Ê
T', then T is finer than T' and T' is coarser than T.
- E is any set, T = P(E) (the discrete topology, the finest topology on E.)
- E is any set, T = {Æ
, E} (the trivial, of indiscrete, topology on E.)
- E is any (non-null) set and p Î
E, T_{p} = {Æ
}È
{X Î
P(E) : p Î
X} (a particular point topology) More to the *ahem* point, E = {0, 1}, T_{0} = {Æ
, {0}, {0, 1}} (the Sherpinkski topology)
- E is any (non-null) set and p Î
E, T_{p} = {E}È
{X Î
P(E) : p Ï
X} (an exculded point topology)
- E is an infinite set, T = {X Î
P(E) : E \ X is infinite} (the Fort topology) Here, for X Ì
E, the complement of X in E (E\X) just means all the elements in E that are not in the subset X; than is, E\X = {x Î
E : x Ï
X}.
- Now, for a topology induced by a metric. Well, you had to know it was coming. ;-) Given a set E and a metric space (E, d), U Í
E is "open relative to metric d" if, "
x Î
U, $
r_{x} Î
R '
V_{d} (x, r_{x}) = {y Î E : d(x, y) < r_{x} } Í
U. Then T_{d} = {S Í
E : S in open relative to d} is the topology induced by d.
A base for a topology T is a subset B Í
T characterized by, "
X Î
T, $
Y_{1}...Y_{n}, Î
B '
È _{i}=1,...n Y_{i} = X. The notion of a base is very nice so that we can talk about a topology without having to write out the whole topology. Suppose you have a set of sets where each set contains a spice; one set for parsley, one for anise, one for sassafras, one for rosemary, etc.... Now, we could mix'n'match our spices to make a topology--make various combinations, intersect/combine those combinations with the other combinations and single spices. Now matter what topology we make--or how big the mess gets in the kitchen--the base will be the collection (set) of singletons containing the original spices. In fact, let E be any set (such as a set of spices) and B be a set of subsets of E (such as a set of spice jars, even if a spice jar contains two or more spices). Suppose B satisfies two conditions: (i) È
B = E and (ii) "
S_{1}, S_{2} Î
B with x Î
S_{1} Ç
S_{2} Î
B, $
S_{3} Î
B '
x Î
S_{3} Í
S_{1} Ç
S_{2}. (In our example, a jar containing oregano and cilantro next to a jar of parsley and rosemary does not necessitate jars for all the separate spices because the intersection is null.) Then there is a unique topology T_{B} on E for which B is a base.
Y Stuff to do in topological spaces
Y Links and Thinks