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spaces in functional analysis, basic concepts related to topological space

terms from the architecture mathematicians seem to lie a lot. This time it's about the concept of space in functional analysis.


One of the most general concepts is that of topological space. It is also in algebraic geometry is applied, not merely "typical Analysis.


We first define the concept of topology . A topology T on a set M is a system of subsets of this set that satisfies certain properties. On the one hand, the amount
M be an element of T, but also the empty set. If two sets in T, so also is their intersection in T are. The union of countably many elements of T will be back in T. Coarse we have said that a set system, with respect to the average education and union as the operation is complete on the set. The attentive reader will see now that T is of average education or association a magma. That is, we have a basic algebraic structure that can start to something.

summarize:

is then (M, T) a topological space . All quantities are the elements of a topology, called open sets .

The complement

a set A is in a different set M defined as follows:

A subset A is called completed if its complement in M is an open set, so this is part of the topology.

Obviously that is, the empty set and M itself are closed sets, although they are open sets, because the complement of the empty set is M, an open set, and the complement of M is the empty set, which is open. In short:

Also, the association two closed sets A and B is again complete. Why? According to the de Morgan laws, the complement of the union of two sets is the average of their Komplementmengen. Both volumes are completed, ie their complements are open. The intersection of two open sets is open. So even the average of the two Komplementmengen is open. Thus the complement of the union of A and B is open.

Similarly, we realize, that the intersection of two closed sets is again closed. The corresponding de Morgan's rule is: The complement of the intersection of two sets is the union of complements of both sets.

The conclusion a subset N M is the average of all closed sets A in M that contain N as a subset.

The interior of a subset N is the union of all open sets in M contained in N as a subset.

A environment of a point m, which is an element in M is a set U, so that there is an open set O that contains m as an element and itself a subset of U. In short:

environments do not have to be open, but, consequently, always include an open environment.