derivation of properties on interior operation

Let X be a topological spaceMathworldPlanetmath and A a subset of X. Then

  1. 1.



    If aint(A), then aU for some open set UA. So aA. ∎

  2. 2.

    int(A) is open.


    Since int(A) is a union of open sets, int(A) is open. ∎

  3. 3.

    int(A) is the largest open set contained in A.


    If U is open set with int(A)UA, then U{VAV open }=int(A), so U=int(A). ∎

  4. 4.

    A is open if and only if A=int(A).


    If A is open, then A is the largest open set contained in A, and so int(A)=A by property 3 above. On the other hand, if int(A)=A, then A is open, since int(A) is, by property 2 above. ∎

  5. 5.



    Since int(A) is open by property 2, int(A)=int(int(A)) by property 4. ∎

  6. 6.

    int(X)=X and int()=.


    This is so because both X and are open sets. ∎

  7. 7.



    (LHS RHS). If aA¯, then aB for every closed setPlanetmathPlanetmath B such that AB. In particular, a(int(A)), for (int(A)) is the complement of an open set by property 2, and A(int(A)) by taking the complement of property 1.

    (RHS LHS). If a(int(A)), then aint(A). If B is a closed set such that AB, then BA. Since B is open, Bint(A) by property 3, so aB, and thus aB. Since B is arbitrary, aA¯ as desired. ∎

  8. 8.



    Set B=A, and apply property 7. So A¯=B¯=(int(B))=int(B)=int(A). ∎

  9. 9.

    AB implies that int(A)int(B).


    This is so because int(A) is open (property 2), contained in A (and therefore contained in B), so contained in int(B), as int(B) is the largest open set contained in B (property 3). ∎

  10. 10.

    int(A)=AA, where A is the boundary of A.


    Recall that A=A¯A¯. So A=A¯(int(A)) by property 7. By direct computation, we have AA=A(A¯(int(A)))=(AA¯)(A(int(A))). Since AA¯= and A(int(A))=A(int(A))=Aint(A), which is int(A) by property 2. ∎

  11. 11.



    Again, by direct computation:

    int(A)A =int(A)(A¯(int(A)))   because A=A¯(int(A))
    =(int(A)A¯)(int(A)(int(A)))    distributes over 
    =A¯X=A¯.   int(A)AA¯

  12. 12.



    By property 11, int(A)Aint(A)=A¯int(A), which, by property 8, is A¯A¯, and the last expression is just X. ∎

  13. 13.



    (LHS RHS). Let C=int(AB). Since C is open and contained in both A and B, C is contained in both int(A) and int(B), since int(A) and int(B) are the largest open sets in A and B respectively. (RHS LHS). Let D=int(A)int(B). So D is open and is a subset of both A and B, hence a subset of AB, and therefore a subset of int(AB), since it is the largest open set contained in AB. ∎

Remark. Using property 7, we see that an alternative definition of interior can be given:

Title derivation of properties on interior operation
Canonical name DerivationOfPropertiesOnInteriorOperation
Date of creation 2013-03-22 17:55:28
Last modified on 2013-03-22 17:55:28
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Derivation
Classification msc 54-00