Description:
Definitions for various forms
Content:
In this note we give the definitions
for forms [1 CE] through
[1 CH],
[67 K] through
[67 T],
[126 G]
through [126 K],
Form 173,
[173 A],
Form 232,
[232 A] through
[232 G],
Form 383, and
[383 A] through
[383 C].
Definition: Let \((X,T)\) be a topological space.
- \(X\) is normal if for every pair of disjoint closed sets
\(A\) and \(B\), there exist disjoint open sets \(C\) and \(D\) such that
\(A\subseteq C\) and \(B\subseteq D\).
- \(X\) satisfies Urysohn's lemma iff it satisfies:
If \(A\), \(B\) are closed and disjoint in \(X\) then there exists a continuous
function \(f:X\rightarrow [0,1]\) separating \(A\) and \(B\). i.e., \(A\subseteq
f^{-1}(0)\) and \(B\subseteq f^{-1}(1).\) (In the remainder of this note
we shall refer to a space that satisfies Urysohn's lemma as a
U space.)
- \(X\) satisfies the Tietze-Urysohn extension theorem
iff it satisfies:
If \(A\) is closed in \(X\) and \(f:A\rightarrow [0,1]\) is a continuous
function, then there exists a continuous extension \(\bar{f}\) of \(f\) to
all of \(X\) taking values in \([0,1]\). (We shall refer to a space
satisfying the Tietze-Urysohn extension theorem as a T space.)
- The disjoint union of the disjoint family \(\{(X_i,T_i)
:i\in k\}\) of topological spaces is the set \(X=\bigcup \{X_i:i\in k\}\) such
that \(O\) is open in \(X\) iff \(O\cap X_i\) is open in \(X_i\) for all \(i\in k\).
- A family \(\cal{K}\) of subsets of a topological space \((X,T)\) is
locally finite (respectively locally countable) iff each
point of \(X\) has a neighborhood meeting only a finite number
(respectively countable number) of elements of \(\cal{K}\).
- \(X\) is paracompact (respectively para-Lindelöf)
iff \(X\) is T\(_2\) and every open cover \(\cal U\) of \(X\) has a locally
finite (respectively locally countable) open refinement
\(\cal V\), (that is, \(\cal V\) is a locally finite open cover
of \(X\) and every member of \(\cal V\) is included in a member of \(\cal U\)).
- A family \(\cal{K}\) of subsets of \(X\) is point finite
(respectively point countable) iff each element of \(X\) belongs
to only finitely many (respectively countably many) members of \(\cal K\).
- \(X\) is metacompact (respectively meta-Lindelöf)
iff each open cover \(\cal U\) of \(X\) has an open point finite
(respectively point countable) refinement.
- A set \(C\subseteq\cal P(X)\) is \(\sigma\)-locally finite
(respectively {\it\(\sigma\)-point finite, \(\sigma\)-disjoint,
\(\sigma\)-discrete}) if \(C = \bigcup_{n\in\omega} C_n\) where each \(C_n\)
is locally finite (respectively, point finite, pairwise disjoint,
discrete).
- Assume that \(\gamma\) is a well ordered cardinal number,
that is, \(\gamma\) is an initial ordinal. A set \(C\subseteq\cal P(X)\)
is \(\gamma\)-locally finite (respectively {\it\(\gamma\)-point
finite, \(\gamma\)-disjoint, \(\gamma\)-discrete}) if
\(C = \bigcup_{\alpha\in\gamma} C_\alpha\) where each \(C_\alpha\) is
locally finite (respectively, point finite, pairwise disjoint, discrete).
- An open cover \(\cal U=\{U_i:i\in k\}\) of \(X\) is shrinkable
iff there exists an open cover \(\cal V=\{V_i:i\in k\}\) of non-empty
sets such that \(\overline{V}_i\subseteq U_i\) for all \(i\in k\). \(\cal V\) is
also called a shrinking of \(\cal U\).
- \(X\) is a PFCS space iff every point finite open cover
of \(X\) is shrinkable.
- \(D\subseteq X\) is discrete if every \(x\in X\) has a
neighborhood \(U\) such that \(|U\cap D|=1\). A set \(C\subseteq\cal P(X)\) is
discrete if \(\forall x\in X\), there is a neighborhood \(U\) of \(x\)
such that \(U\cap A\ne\emptyset\) for at most one element \(A\in C\). (Thus,
\(D\subseteq X\) is discrete iff \(\{\{d\}: d\in D\}\) is discrete.)
- \(X\) is cwH (collectionwise Hausdorff) iff if \(X\) is T\(_2\)
and for every closed and discrete subset \(D\) of \(X\) there exists a
family \(U=\{U_d:d\in D\}\) of disjoint open sets such that \(d\in U_d\)
for all \(d\in D\).
- \(X\) is cwH(\(B\)) (collectionwise Hausdorff with respect to
the base \(B\)) iff if \(X\) is T\(_2\)
and for every closed and discrete subset \(D\) of \(X\) there exists a
family \(U=\{U_d:d\in D\}\) of disjoint open sets such that \(U\subseteq B\)
and \(d\in U_d\) for all \(d\in D\).
- \(X\) is cwN (collectionwise normal) if \(X\) is \(T_2\) and
for every discrete \(C\) of pairwise disjoint closed subsets of \(X\),
there exists a family \(U=\{U_A:A\in C\}\) of disjoint open
sets such that \(A\subseteq U_A\) for all \(A\in C\). (cwN(\(B\)) is defined
similarly to cwH(\(B\)) in 15.)
Howard-Rubin number:
141
Type:
Definitions
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