topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Basic homotopy theory



For topological spaces

For SXS \subset X a subset of a topological space XX, a, interior point of SS is a point xSx \in S which has a neighbourhood in XX that is contained in SS. The union of all interior points is the interior S S^\circ of SS. It can be defined as the largest open set contained in SS.

In general, we have S SS^\circ \subseteq S. SS is open if and only if S =SS^\circ = S, so that in particular S =S S^{\circ\circ} = S^\circ. This makes the interior operator P(X)P(X):SS P(X) \to P(X): S \mapsto S^\circ a co-closure operator. It also satisfies the equations (ST) =S T (S \cap T)^\circ = S^\circ \cap T^\circ and X =XX^\circ = X. Moreover, any co-closure operator cc on P(X)P(X) that preserves finite intersections must be the interior operation for some topology, namely the family consisting of fixed points of cc; this gives one of many equivalent ways to define a topological space.

Compare the topological closure S¯\bar{S} and frontier S=S¯S \partial S = \bar{S} \setminus S^\circ.

For toposes

The interior of a subtopos j\mathcal{E}_j of a Grothendieck topos \mathcal{E}, as well as the exterior, were defined in an exercise in SGA4: Int( j)Int(\mathcal{E}_j) as the largest open subtopos contained in j\mathcal{E}_j. The boundary of a subtopos is then naturally defined as the subtopos complementary to the (open) join of the exterior and interior subtoposes in the lattice of subtoposes.


Relation to topological closure


Let (X,τ)(X,\tau) be a topological space and let SXS \subset X be a subset. Then the topological interior of SS equals the complement of the topological closure Cl(X\S)Cl(X\backslash S) of the complement of SS:

Int(S)=X\Cl(X\S). Int(S) = X \backslash Cl\left( X \backslash S \right) \,.

By taking complements once more, the statement is equivalent to

X\Int(S)=Cl(X\S). X \backslash Int(S) = Cl( X \backslash S ) \,.

Now we compute:

X\Int(S) =X\(UopenUSU) =USX\U =CclosedCX\SC =Cl(X\S) \begin{aligned} X \backslash Int(S) & = X \backslash \left( \underset{{U \, open} \atop {U \subset S}}{\cup}U \right) \\ & = \underset{U \subset S}{\cap} X \backslash U \\ & = \underset{{C\, closed} \atop {C \supset X \backslash S}}{\cap} C \\ & = Cl(X \backslash S) \end{aligned}


Revised on May 9, 2017 13:11:41 by Urs Schreiber (