Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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2 I Fundamental Concepts<br />
C1 /0,X ∈ C;<br />
C2 if {Uα | α = 1,...,n} is any f<strong>in</strong>ite set of elements of C,then<br />
n�<br />
Uα ∈ C<br />
α=1<br />
(<strong>in</strong> other words, any f<strong>in</strong>ite union of closed sets is a closed set);<br />
C3 if {Uα | α ∈ J} is any set of elements of C,then<br />
�<br />
Uα ∈ C<br />
α∈J<br />
(<strong>in</strong> other words, any <strong>in</strong>tersection of closed sets is a closed set).<br />
For any space X,theclosure Y of a subset Y ⊂ X is the <strong>in</strong>tersection of all closed<br />
subsets of X which conta<strong>in</strong> Y; it is, obviously, the smallest closed subset conta<strong>in</strong><strong>in</strong>g<br />
Y, andY∈C. The<strong>in</strong>terior ˚Y of Y is the union of all open subsets of X conta<strong>in</strong>ed<br />
<strong>in</strong> Y ; it is the largest open set conta<strong>in</strong>ed <strong>in</strong> Y ,and˚Y ∈ U. The follow<strong>in</strong>g lemma is a<br />
useful characterization of the closure of a set.<br />
(I.1.1) Lemma. Let Y ⊂ X; theny∈ Y if and only if every open set U conta<strong>in</strong><strong>in</strong>g y<br />
<strong>in</strong>tersects Y .<br />
Proof. In fact, if U ∩Y = /0, then X �U is a closed subset of X which conta<strong>in</strong>s Y ;<br />
s<strong>in</strong>ce y belongs to the closure of Y, it follows that y ∈ X �U, a contradiction.<br />
Conversely, suppose that there exists a closed set C ⊂ X conta<strong>in</strong><strong>in</strong>g Y and which<br />
does not conta<strong>in</strong> y. ThenX �C is an open set, y ∈ X �C, and(X �C) ∩Y = /0, a<br />
contradiction. �<br />
For any set X, there are two topologies that come immediately to m<strong>in</strong>d:<br />
1. The discrete topology whose open sets are all the subsets of X, thatistosay,<br />
Ud = P(X)=2X .<br />
2. The trivial topology with Ub = {/0,X}.<br />
Clearly, Ud ⊃ Ub and this fact br<strong>in</strong>gs to m<strong>in</strong>d the idea of compar<strong>in</strong>g two topologies<br />
on the same set: we shall say that topology U ′ is f<strong>in</strong>er than topology U if U ′ ⊃ U;<br />
hence, the discrete topology of X is f<strong>in</strong>er than the trivial one.<br />
AsetBof subsets of X is a basis for a topology on X or a basis of open sets for<br />
X if:<br />
B1 For each x ∈ X,thereisaB∈Bsuch that x ∈ B (that is to say, X = �<br />
B∈B B).<br />
B2 If B1,B2 ∈ B and B1 ∩B2 �= /0, then for each x ∈ B1 ∩B2 there exists B3 ∈ B<br />
such that x ∈ B3 ⊂ B1 ∩ B2.<br />
AbasisBgenerates a topology U on X automatically, by requir<strong>in</strong>g that U ⊂ X<br />
be open <strong>in</strong> the topology U if, and only if, for each x ∈ U, there exists B ∈ U such that<br />
x ∈ B ⊂ U; formally,<br />
U ∈ U ⇐⇒ (∀x ∈ U)(∃B ∈ B) x ∈ B ⊂ U.