Cellularity

Cellularity ( Suslin number ) is a topological characteristic of a topological space $X$ ${\ displaystyle X}$ $X$ determined by the maximum number of open pairwise disjoint sets from $X$ ${\ displaystyle X}$ $X$ . It is a cardinal invariant and is denoted by $c(X)$ ${\ displaystyle c (X)}$ ${\ displaystyle c (X)}$ .

As with many general topological invariants, finite cellularity is not of interest; it is considered that it is no less than countable (i.e. $\aleph _{0}$ ${\ displaystyle \ aleph _ {0}}$ $\ aleph_0$ )

Heredity

Not a hereditary invariant , i.e. a subspace $U\subseteq X$ ${\ displaystyle U \ subseteq X}$ $U\subseteq X$ may have greater cellularity than $c(X)$ ${\ displaystyle c (X)}$ $c(X)$ . For example, a point is enough $0$ ${\ displaystyle 0}$ $0$ in the segment $[0,1]$ ${\ displaystyle [0,1]}$ $[0,1]$ multiply an uncountable number of times, then the subspace of multiplied zeros will have greater cellularity than the segment, that is, more $\aleph _{0}$ ${\ displaystyle \ aleph _ {0}}$ $\aleph_0$ , i.e $\aleph _{0}=c(X)<hc(X)={\mathfrak {c}}$ ${\ displaystyle \ aleph _ {0} = c (X) <hc (X) = {\ mathfrak {c}}}$ $\aleph _{0}=c(X)<hc(X)={\mathfrak {c}}$ . Another example of non-inheritance of cellularity is the Nemytsky plane .

Relationship with other invariants

The cellularity of space does not exceed its density (which, in turn, does not exceed weight ): $c(X)\leqslant d(X)\leqslant w(X)$ ${\ displaystyle c (X) \ leqslant d (X) \ leqslant w (X)}$ $c(X)\leqslant d(X)\leqslant w(X)$ . Also, cellularity does not exceed the spread (which also does not exceed the weight): $c(X)\leqslant hc(X)\leqslant w(X)$ ${\ displaystyle c (X) \ leqslant hc (X) \ leqslant w (X)}$ $c(X)\leqslant hc(X)\leqslant w(X)$ .

For linearly ordered spaces, their character does not exceed cellularity: $\chi (X)\leqslant c(X)$ ${\ displaystyle \ chi (X) \ leqslant c (X)}$ $\chi (X)\leqslant c(X)$ . In addition, for linearly ordered spaces, cellularity coincides with the spread and the Lindelöf hereditary number : $c(X)=hc(X)=hl(X)$ ${\ displaystyle c (X) = hc (X) = hl (X)}$ $c(X)=hc(X)=hl(X)$ .

Do not exceed the cellularity of the topological space $X$ ${\ displaystyle X}$ $X$ its Lindelöf number and extent (in turn, not exceeding the Lindelöf number): $e(X)\leqslant l(X)\leqslant c(X)$ ${\ displaystyle e (X) \ leqslant l (X) \ leqslant c (X)}$ $e(X)\leqslant l(X)\leqslant c(X)$ .

Examples

For the real line $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ $\mathbb {R}$ : $c(\mathbb {R} )=\aleph _{0}$ ${\ displaystyle c (\ mathbb {R}) = \ aleph _ {0}}$ $c(\mathbb {R} )=\aleph _{0}$ . For natural and integer numbers: $c(\mathbb {Z} )=c(\mathbb {N} )=\aleph _{0}$ ${\ displaystyle c (\ mathbb {Z}) = c (\ mathbb {N}) = \ aleph _ {0}}$ $c(\mathbb {Z} )=c(\mathbb {N} )=\aleph _{0}$ .

For discrete power space $\tau$ ${\ displaystyle \ tau}$ $\tau$ : $c(D_{\tau })=\tau$ ${\ displaystyle c (D _ {\ tau}) = \ tau}$ $c(D_{\tau })=\tau$ .

For a hedgehog prickly $\tau$ ${\ displaystyle \ tau}$ $\tau$ : $c(J(\tau ))=\tau$ {\ displaystyle c (J (\ tau)) = \ tau} $c(J(\tau ))=\tau$ . (When $\tau \geqslant \aleph _{0}$ ${\ displaystyle \ tau \ geqslant \ aleph _ {0}}$ $\tau \geqslant \aleph _{0}$ (it’s enough to take an open set in each “needle” that does not go beyond the “needle”).

In general for subspace $U$ ${\ displaystyle U}$ from Euclidean space $\mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ : $c(U)\leqslant \aleph _{0}$ ${\ displaystyle c (U) \ leqslant \ aleph _ {0}}$ .

Literature

Engelking, Ryszard. General topology. - M .: Mir , 1986 .-- S. 103.333. - 752 s.

Cellularity

Heredity

Relationship with other invariants

Examples

Literature

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