Let's get our Topology basics fundementaly comfortable by studying at the Genesis of our project...

http://neil-strickland.staff.shef.ac.uk/Wurble.html

And here is Geometery explained a little more in depth...us2u

http://library.thinkquest.org/2647/geometry/geometry.htm

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Geometric introduction to Topology


http://fds.oup.com/www.oup.co.uk/pdf/0-19-851597-9.pdf

Splitting off rational parts in
homotopy types

http://www.math.kyushu-u.ac.jp/~iwase/Talks/lec04e-beijing.pdf

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quote


This paper is an enlargement of the first one  I wrote which had to see with a topological model for the structure of elementary particles of physics according to the Ghassemi’s space-time-mass theory of the universe . From that paper we can establish the following problem:



¿Which ones should be the last and fundamental building blocks from which matter is made taking into account a topological model and according to the Ghassemi’s space-time-mass theory of the universe?



The answer to this problem was already established in the same one as it follows:



There are only two kinds of elementary particles being the building blocks of space-time: the first one, whose topological structure has positive gaussian curvature and the second one whose topological structure has negative gaussian curvature.



With this hypothesis it was possible to explain matter creation, electrostatic interaction of charged particles, etc.



http://www.space-time-mass.com/totalpaper.htm


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http://www.math.niu.edu/~rusin/known-math/index/54-XX.html


[ At link..] Introduction ~

Topology is the study of sets on which one has a notion of "closeness" -- enough to decide which functions defined on it are continuous. Thus it is a kind of generalized geometry (we are still interested in spheres and cubes, for example, but we might consider them to be "the same", yet distinct from a bicycle tire, which has a "hole") or a kind of generalized analysis (we might think of the functions f(x)=x^2 and f(x)=|x| as being "the same", and yet distinct from f(x)=signum(x)=x/|x|, which has a discontinuity).

More formally, a topological space is a set X on which we have a topology -- a collection of subsets of X which we call the "open" subsets of X. The only requirements are that both X itself and the empty subset must be among the open sets, that all unions of open sets are open, and that the intersection of two open sets be open.

This definition is arranged to meet the intent of the opening paragraph. However, stated in this generality, topological spaces can be quite bizarre; for example, in most other disciplines of mathematics, the only topologies on finite sets are the discrete topologies (all subsets are open), but the definition permits many others. Thus a general theme in topology is to test the extent to which the axioms force the kind of structure one expects to use and then, as appropriate, introduce other axioms so as to better match the intended application.

For example, a single point need not be a closed set in a topology. Does this seem "inappropriate"? Then perhaps you are envisioning a special kind of topological space, say a a metric space. This alone still need not imply the space looks enough like the shapes you may have seen in a textbook; if you really prefer to understand those shapes, you need to add the axioms of a manifold, perhaps. Many such levels of generality are possible.

Since the axioms of topology are stated in terms of subsets of X, it should be no surprise that one branch of topology is closely related to set theory, particularly "descrïptive set theory". Here one considers general constructs such as closures of sets, limits, convergence, and nets. One can look at topologies related to order or cardinality, and so create extraordinarily large topological spaces. By using the axiom of choice, one may prove the existence of topological spaces with peculiar properties. In particular, there are questions about topology which can be reduced to questions of set theory, whose answer then depends on the axioms of set theory chosen.

As in other branches of axiomatic mathematics, we may (in category-theoretic style) make some basic constructs. The most important functions between topological spaces are the continuous ones (a definition borrowed from analysis), which we use to define homeomorphisms -- functions which can be used to demonstrate that two spaces are "the same". We can define products of spaces (and coproducts -- unions), subspaces, quotient spaces, and so on. On the one hand, each is a "universal" solution to some problem which can be stated in terms of the existence of maps and spaces related via commutative diagrams; this aspect makes them useful tools for algebraic topology. On the other hand, each can be studied (and generalized) internally, making them useful tools for analysis (including semicontinuous functions for example).

In order to get more significant results, one must restrict to spaces with some additional properties. The precise set of additional axioms depends on the intended results. For example, if we would like to know which spaces might have a topology which is consistent with a metric, we know individual points must be closed; the axiom that this is true (the "T_1" axiom) is but one of a number of separation axioms. A great deal of work has been done to see the independence of these axioms, the extent to which they are preserved under the constructions of the previous paragraph, and so on. The same is true of other types of axioms designed to focus attention on "well-behaved" spaces. For example, there are cardinality axioms (e.g., metric spaces have the additional property that the topology at a point is countable), compactness axioms (e.g. a space would have to be locally compact to be a manifold), connectedness axioms, and so on. In each case, there are a number of choices for how tight the axioms should be: is one interested in weak conclusions about a large family of spaces, or stronger conclusions about a family of more particular interest? Well-known results concerning these properties include a version of the Baire Category Theorem (nowhere dense subsets), Tychonoff's theorem (products of compact spaces), Urysohn's Lemma and Tietze's Theorem (functions on well-separated spaces), and compactness criteria (Bolzano-Weierstrass, Heine-Borel).



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