Difference between revisions of "List of topological properties"

From Maths
Jump to: navigation, search
(Added closure)
(Added interior - done quickly)
 
Line 43: Line 43:
 
{{XXX|Tidy this up}}
 
{{XXX|Tidy this up}}
 
|  
 
|  
 +
|-
 +
! {{link|Interior|topology}}
 +
| {{MM|\text{Int}(A,X):\eq\bigcup_{U\in\{V\in\mathcal{J}\ \vert\ V\subseteq A\} }U}}{{rITTMJML}}
 +
|
 +
| Could be union of all interior points, see [https://wiki.unifiedmathematics.com/index.php?title=Interior&oldid=1412 here]
 
|-
 
|-
 
! rowspan="1" | {{link|Interior point|topology}}
 
! rowspan="1" | {{link|Interior point|topology}}

Latest revision as of 19:33, 16 February 2017

Stub grade: A*
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Needs linking in to places. Because density is SPRAWLED all over the place right now

Index

Here (X,J) is a topological space or (X,d) is a metric space in the definitions.

Property Topological version Metric spaces version Comments
Closure Let AP(X) be given. The closure of A, denoted ¯A is defined as follows:
  • ¯A:={CC(X) | AC}[1] - where C(X) denotes the set of closed sets of X

Informally, it is the smallest closed set containing A.

  • Note that the largest closed set c
Probably something with limit points See also:
Dense set For AP(X) we say A is dense in X if:
  • UJ[UA][1][Note 1]
For AP(X) we say A is dense in X if:
  • xXϵ>0[Bϵ(x)A][1]

Caveat:This is given as equiv to density by[1] - also obviously follows from it!

See also:
Equivalent statements
The following are equivalent to the definition above.
  1. Closure(A)=X[1]
  2. XA contains no (non-empty) open subsets of X[1]
    • Symbolically: UJ[U, which we can easily manipulate to get: \forall U\in\mathcal{J}\exists p\in U[p\notin X-M]
  3. X-A has no interior points[1] (see below)
    • Symbolically we may write this as: \forall p\in X-A\left[\neg\left(\exists U\in\mathcal{J}[p\in U\wedge U\subseteq A\right)\right]
      \iff\forall p\in X-A\forall U\in\mathcal{J}[\neg(p\in U\wedge U\subseteq A)]
      \iff\forall p\in X-A\forall U\in\mathcal{J}[(\neg(p\in U))\vee(\neg(U\subseteq A))] - by the negation of logical and
      \iff\forall p\in X-A\forall U\in\mathcal{J}[p\notin U\vee U\nsubseteq A] - of course by the implies-subset relation we see (A\subseteq B)\iff(\forall a\in A[a\in B]), thus:
      \iff\forall p\in X-A\forall U\in\mathcal{J}\big[p\notin U\vee(\exists q\in U[q\notin A])\big]
TODO: Tidy this up
Interior \text{Int}(A,X):\eq\bigcup_{U\in\{V\in\mathcal{J}\ \vert\ V\subseteq A\} }U[2] Could be union of all interior points, see here
Interior point For a set A\in\mathcal{P}(X) and a\in A, a is an interior point of A if:
  • \exists U\in\mathcal{J}[a\in U\wedge U\subseteq A][1]
For a set A\in\mathcal{P}(X) and a\in A, a is an interior point of A if:
  • \exists\epsilon>0[B_\epsilon(a)\subseteq A][1]

Caveat:Basically follows from topological definition, these are closely related

Notes

  1. Jump up There are a few simple equivalent conditions, any of these may be the definition given in a book, although \text{Closure}(A)\eq X is quite common

References

  1. Jump up to: 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha
  2. Jump up Introduction to Topological Manifolds - John M. Lee