Our aim of topology on a set to contrust open sets and the idea of contrust of open sets is in stanrdad topology on real number (metric topology). We know that any open set is arbitary uniom of intervals.

I suppose it's easier to grasp the idea behind an axioms of closure of Kuratowski. Then original axioms of topology became an observation of properties of open sets which are equivalent to axioms of closure.

But now people tend to use axioms of open sets since they are better for proving and defining things (try to define compactness in terms of closure operator). Also they're better for defining topologies on sets (in terms of basis) since you can effectively describe algebra of open sets by using generators of this algebra.

Dear Gauree

Topology ( The non quantum geometry)

Imagine that you have drawn a circle inside a plane made of rubber , and you should follow the following rules :

you can expand , stretch , contract this plane but without tearing it ,

after you do this action , there are some properties of the circle preserved and

others are lost, of course the circle may turn into triangle or another shape ( which is equivalent to the circle ) it is similar to a closed path,the preserved properties are called topological properties and the lost are not ,as an example : the inside point to the circle stay inside ( it topological property) ,

the boundary point stay boundary point (it is topological property)

but we lost the area ( area is not topological property)

Of course to define topological spaces we use the ( the 3 - axioms

concerning the open sets )

but concerning your question which concentrate on the geometric point of view ,

we can topologize the above plane by assuming that each region inside the circle

(or any closed path analogue to circle) without the boundary is an open set .

For consistency we need the intersection of any family of open sets stay open and

also the union of any open family stay open .

In case of the union of infinite open sets there is nothing to worry about ,

this union will stay open but unfortunately in case of infinite intersections

this intersection may turn into ( non open set )

As an example : the intersection of concentric circles with radii (1/n) as n goes to infinity the intersections is not open , in fact it is the centre ( one point ,

closed set which is not open ),so we need to avoid such non preferable case,

therefore we restrict our definition or (the axioms) to exclude this case of infinite

intersections into the finite case .

Hope this answered your question .

regards

I.Kaddoura

I wrote a partial answer to this question on mathoverflow: http://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets/19156#19156

The idea of a topology generalizing the notion of a metric might be helpful here. In a metric space, the set of points whose distance from a fixed point P is less than epsilon, epsilon greater than 0, is an open set. If we insist on allowing arbitrary intersections of open sets, then every point would be an open set.

@David

Did you read the previous answers ?

best

Yes I have read all the answers. Now my doubts are almost clear.

I think the open-set definition of topological space is unintuitive at first glance, especially to someone who hasn't seen metric spaces. However, I think the definition of metric on a set is natural and well-motivated, as is the definition of topological base: just a bunch of distinguished neighborhoods; a neighborhood of a point should be "puffy," enough so that the overlap of two neighborhoods contains a neighborhood of each of its points. Of course each point should have at least one neighborhood; and if we define a set to be open just in case each of its points has a neighborhood contained in the set (so each element of the open set is "well within" the set), then the fact that the collection of open sets is closed under finite intersections and arbitrary unions is a consequence of the more natural definition. By the way, there is the important study of P-spaces: countable intersections of open sets are open. A good reference is "Rings of Continuous Functions," by Gillman and Jerison; also a fine book is Walker's "The Stone-Cech Compacification."

Originally topology grew out of analysis, in which the notion of continuity was the metric notion given by the familiar "epsilon and delta" definition. The problem with this (seen retrospectively through the lens of category theory) is that continuous maps are the wrong maps for metric spaces -- they don't preserve the structure -- the "right" maps for metric spaces are isometric embeddings and isometries. So one would like to know what structure the continuous maps are preserving.

A little fiddling with the epsilon and delta definition makes clear that the continuous maps between metric spaces are precisely the maps which preserve, under inverse image, the family of sets which are arbitrary unions of metric balls. In any metric space this family of sets has the familiar properties in the axioms of a topology: it contains the empty set (as the union of the empty family of metric balls), the whole space, is closed under arbitrary unions, but only under finite intersections (proof of the last being a standard exercise in elementary topology courses).

What the continuous functions between two metric spaces are depends only on this family of sets such defined by each (now called the metric topology), and thus, different metrics become equivalent when they define the same metric topology (if one is only interested in studying continuous functions, rather than using the metric to define distance for some other purpose). It is then a short, but important, step to throw away the metric(s) entirely and study spaces equipped with a topology, not necessarily induced by a metric.

Maybe, for the sake of an "intuitive" comprehension, we can approach this by changing our point of view. Think of a Topology as a rational system of subsets defined for an ultimate purpose: give to some set X the possibility for constructing the notion of "proximity", "continuity", and others relative notions derived from the revolutionary idea of Limit. According to this approach, it turns out that Topology is a generalization of real analysis...

Now... As a "rational system", we need to clarify the construction rules for all sets of this topology; or any other topology defined over X...

These two statements (arbitrary unions and finite intersections) tell us how do we produce all the open sets belonging to our Topology.

Now... Unions and intersections are our "generating machines" of open sets in our topology. But, although it seems not to be a problem to allow arbitrary unions to be open sets, we can mess up assuming the same for arbitrary intersections. And the reason is plain and simple: If we allow in our "generating machines" arbitrary intersections, all Topologies we can define will be no more than a boring discrete topology