Talk:Boundary (topology)

message: I'm wondering about the diagram illustrating the relationship between points in S, limit points (A) and boundary points (B), along with isolated and interior points.  Is this diagram only about subsets of R^n  (the bullet just above the diagram references R^n , although it isn't clear that the diagram only pertains to this point) with usual ball topology ? Otherwise there are topological spaces where there exist isolated points which aren't boundary points, no ?  Take X with a single point "x" along with its trivial topology and consider the subset S=X itself: "x" is an isolated point, but not a boundary point; the boundary of X is empty.  
Similarly, an interior point is not necessarily a limit point, for example if X has the discrete topology, since singletons are open sets then for every point "s" of a subset S there exists neighborhoods completely contained in S, hence all points of S are interior. Those same points are not limit points but isolated points (since these singletons sets are also neighborhoods of points "s" containing only "s".) 70.81.15.136 (talk) 02:37, 27 February 2008 (UTC)
replies: text: I have added that S must be a subset of R^n , now the text should be correct. But I don't find this diagram very interesting nor particularly clear, in my opinion it could be removed. Daniele 84.220.23.101 (talk) 19:49, 1 March 2008 (UTC)
thread_title:  Venn Diagram 

message: Two media may be located so close to one another that there is no third medium between the two. Nevertheless there is a boundary between them except that the boundary is Nothingness which is 'something' but not space or time. An observer wanting to cross from one medium to the other has to cross the boundary. Change of one medium for the other creates the duality of exitence non-existence because in the boundary one medium ends existence and the other starts. If the boundary is perfect the chasnge is quantittive. If it is imperfect, and it has magnitude either spatial or temporsal, there is transformation through the boundary. There is no observation in the boundary if it cannot accommodate observer's 'now' containing consciousness. Two units 'now' are required to observe a difference. It is the boundary and not 'now' that contains change from existence to non-existence and the other way round. Sleep is the boundary in whcih we are not conscious of existence of anything. 
     There is only one medium common for all the space times. It is the observer who creates from this medium different space times and boundaries between them. The medium in which creation takes place is the Nothingness which is neither space nor time. Without space time only the Nothingness is left as the boundary between the existence and non-existence of the observer's 'self'. A very important boundary is the one between the material and the immaterial space times. Human observer, located in the material world uses 'now' of constant magnitude. The result is that all forms of change in his world are measured with that static unit 'now', including changes in that unit 'now' itself. Each different space time uses a different unit 'now'. To change his location from the material world to the immaterial, the observer can use the continuous method when he changes his unit 'now' until it is small enough to cross the boundary. The other way is to change the unit 'now' quantitatively. In either case the observer has to change from existence to non-existence in the material world before he can exist in the immaterial space time. KK (78.146.162.196 (talk) 11:49, 3 April 2010 (UTC)
thread_title:  The boundary 

message: Let $n = 1$, so consider the real line. Let $A = [-1,1]$ be the unit closed circle. A is closed. Suppose it is the boundary of some set of the reals, say $B$. Then $B - A$ is open. Furthermore, it is the boundary of $X = \mathbb R - B$, which is open too. Hence $X$ consist of two open components, say $O1, O2$. But as $Y = {x real, x > 1}$ is open, both $Y \cap O1$, $Y \cap O2$ are open. Furthermore, as $X = O1 \cup O2$, the union of these two open intersections is the whole of Y. But Y is connected. Contradiction. 
Hence the last stated property, that each closed set of $R^n$ is the boundary of some subset of $R^n$ is clearly wrong.
131.220.99.58 (talk) 14:07, 9 July 2010 (UTC)
thread_title:  Dubious property