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\topmatter
\title
Definitions (\revised)
\endtitle
\endtopmatter
%\rightheadtext{Definitions}
%\leftheadtext{Pavel Pyrih}
\vskip 1in
\document

% absolute end point

\demo{ Definition}
A point $p$ of a continuum $X$ is called an {\bf absolute
end point} of $X$ if $X \setminus \{p\}$ is a composant of
$X$.
\enddemo

% absolutely terminal continuum

\demo{ Definition}
A proper subcontinuum $K$ of a continuum $X$  is said
to be an  {\bf absolutely terminal continuum} of $X$
provided that $K$ is a terminal continuum of
each subcontinuum $L$  of $X$ which properly contains $K$.
\enddemo


% almost chainable

\demo{ Definition}
A continuum $M$ is {\bf almost chainable} if, for each positive
number $\eps$, there exists an open cover $D$ of $M$ with
mesh $\eps$ and a chain $C$ of elements of $D$ and an end
link $L$ of $C$ such that no element of $D \setminus C$
intersects any link of $C$ other than $L$ and every point of $M$
is at a distance less than $\eps$ from some link of $C$.
\enddemo

% aposyndetic

\demo{ Definition}
A connected space $X$ is {\bf aposyndetic} at $H$ with respect to
$K$ if there is a closed connected subset of $X$ with $H$ in its
interior and not intersecting $K$, and $X$ is {\bf aposyndetic}
if it is aposyndetic at each point with respect to every other
point.
\enddemo


% arc

\demo{ Definition}
An  {\bf arc} is any space which is homeomorphic to the
closed interval $[0,1]$.
\enddemo

% atomic

\demo{ Definition}
If $f$ is a mapping of a continuum $X$ onto a continuum $Y$ and,
for each subcontinuum $K$ of $X$ such that $f(K)$ is
nondegenerate $K = f^{-1}\( f\(K\) \)$, then $f$ is {\bf atomic}.
Atomic maps are monotone.
\enddemo

% atriodic

\demo{ Definition}
A continuum that contains no subcontinuum which is itself a
triod is said to be {\bf atriodic}
\enddemo

%  biconnected

\demo{ Definition}
A space is {\bf biconnected} if it is not the sum of two
mutually exclusive nondegenerate connected point sets.
\enddemo

% bihomogeneous

\demo{ Definition}
Suppose $X$ is a space such that, for each two points $a$
and $b$, there is a homeomorphism of $X$ onto itself such
that $h(a) = b$ and $h(b) = a$.  Then $X$ is {\bf
bihomogeneous}].
\enddemo


% Borel sets


\demo{ Definition}
The family $\Cal F$ of {\bf Borel sets} of a space $S$ is the
smallest family satisfying the conditions:
\roster
\item  Every closed set belongs to $\Cal F$;
\item  If $X$ is in $\Cal F$, then $S \setminus X$ is in $\Cal
F$; and
\item  The countable intersection of elements of $\Cal F$ belongs
to $\Cal F$.
\endroster
\enddemo

% branch point

\demo{ Definition}
A {\bf branch point} of a continuum is the vertex of a simple
triod lying in that continuum.
\enddemo



% chainable

\demo{ Definition}
A {\bf simple chain} is a finite sequence $L_1, L_2, \dots,L_n$
of open sets such that $L_i$ intersects $L_j$ if and only if
$|i-j| \leq 1$.  The terms of the sequence $L_1, L_2, \dots,L_n$
are called the {\bf links} of the chain.
An {\bf $\eps$-chain} is a chain each of whose links has
diameter less than $\eps$.  A continuum $M$ is {\bf
chainable} if, for each positive number $\eps$, $M$ can be
covered by an $\eps$-chain.
\enddemo

% circle

\demo{ Definition}
A  {\bf circle} is any space which is homeomorphic to the
unit circle.
\enddemo

% composant

\demo{ Definition}
If a point $p$ of a continuum $X$ is given, then the {\bf
composant of $X$ belonging to } $p$ is defined to be the
union of all proper subcontinua of $X$ which contain $p$. A
set is a {\bf composant of } $X$ provided it is the
composant of $X$ belonging to $p$ for some point $p$ of $X$.
\enddemo



%  confluent

\demo{ Definition}
A mapping $f$ of a compact space $X$ onto a compact space $Y$ is
{\bf confluent} if, for each continuum $C$ in $Y$, each component
of $f^{-1}(C)$ is mapped onto $C$ by $f$.
\enddemo

% connected im kleinen

\demo{ Definition}
A space is {\bf connected im kleinen} if it is aposyndetic at
each point with respect to each closed point set not containing
that point.
\enddemo


% converges homeomorphically to the continuum

\demo {Definition}
The statement that the sequence $M_1, M_2, \dots$ {\bf converges
homeomorphically to the continuum $M$} means there exists a
sequence $h_1, h_2, \dots$ of homeomorphisms such that, for
each positive integer $i$, $h_i$ is a homeomorphism from $M_i$
onto $M$ and for each positive number $\eps$ there exists a
positive integer $N$ such that if $j > N$ then, for all $x$,
dist$(h_j(x),x) < \eps$.
\enddemo

% continuum

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

% completely regular

\demo{ Definition}
A continuum is {\bf completely regular} if each nondegenerate
subcontinuum has a nonempty interior.
\enddemo

% curve

\demo{ Definition}
A  {\bf curve} is any one-dimensional continuum.
\enddemo


% cut point

\demo{ Definition}
Let $S,T)$ be a connected topological space,
and let $p \in S$. If $S \setminus \{p\}$ is connected,
then $p$ is called a {\bf non-cut point of} $S$.
If $S \setminus \{p\}$ is not connected,
then $p$ is called a {\bf cut point of} $S$.
\enddemo




% cyclic element

\demo{ Definition}
If $M$ is a semi-locally connected continuum, a {\bf cyclic
element} of $M$ is either a cut point of $M$, an end point
of $M$, or a nondegenerate subset of $M$ which is maximal
with respect to being a connected subset of $M$ without cut
points.
\enddemo


% decomposable

\demo{ Definition}
A continuum $X$ is said to be  {\bf decomposable} provided
that $X$ can be written as the union of two proper
subcontinua.
\enddemo




% $\delta$-connected

\demo{ Definition}
A continuum is {\bf $\delta$-connected} if each two of its
points can be connected by an hereditarily decomposable
irreducible subcontinuum.
\enddemo

% dendroid

\demo{ Definition}
A {\bf dendroid} is an arcwise connected, hereditarily unicoherent continuum.
\enddemo




%  dispersion point

\demo{ Definition}
A point $p$ of a nondegenerate connected space $X$ is an {\bf
explosion point}, or {\bf dispersion point}, of $X$ provided that
$X \setminus \{p\}$ is totally disconnected.
\enddemo

% end point

\demo{ Definition}
A point $p$ of a continuum $X$ is called an {\bf end point}
of $X$ if for each two subcontinua of $X$ both containing
$p$, one of the subcontinua contains the other.
\enddemo

% fan

\demo{ Definition}
A {\bf fan} is a dendroid with exactly one ramification point.
\enddemo

% $\Cal F$-equivalent

\demo{ Definition}
If $X$ and $Y$ are continua, we say that $Y$ is
{\bf $\Cal F$-equivalent} to $X$ provided there is a mapping in
$\Cal F$ from $X$ onto $Y$ and a mapping in $\Cal F$ from $Y$
onto $X$.
\enddemo




%  finitely Suslinean

\demo{ Definition}
A continuum $X$ is {\bf finitely Suslinean} if, for each positive
number $\eps$, $X$ does not contain infinitely many mutually
exclusive subcontinua of diameter greater than $\eps$.
\enddemo

% finitely linear

\demo{ Definition}
A function $f\: I \to Y$, where $I$ is a closed interval of the
real line, is called {\bf finitely linear} provided there exists
a positive integer $m$ such that $I$ can be decomposed, for each
$\eps > 0$, into a finite number of closed subintervals
$I_1, I_2, \dots, I_k$ each of length less than $\eps$ and
with the property that the set $f(I_i)$ meets at most $m$ of the
sets $f(I_1), f(I_2), \dots, f(I_k)$ for
$i = 1, 2, \dots, k$.\enddemo

% fixed point property

\demo{ Definition}
A topological space $X$ is said to have the {\bf fixed point property} provided
that for every mapping $f:X\to X$ there exists $x\in X$ such that $f(x)=x$.
\enddemo

% fixed set property

\demo{ Definition}
A topological space $X$ is said to have the {\bf fixed set
property} for a certain class $\Cal C$ of maps of $X$ onto itself
provided there exists, for each non-empty closed set $A$ in $X$,
a map $f$ in $\Cal C$ such that $f(x) = x$ if and only if $x$ is
in $A$.
\enddemo


% free arc

\demo{ Definition}
A {\bf free arc} in $X$ is any subset of $X$ of the  form
$A \setminus \{x,y\}$ where $A$ is an arc in $X$ with end points $x$
and $y$ and $A \setminus \{x,y\}$ is open in $X$.
\enddemo


% graph

\demo{ Definition}
A {\bf graph} is a continuum which can be written as the
union of finely many arcs any two of which are either
disjoint or intersect only in one or both their endpoints.
\enddemo

% half-ray curve

\demo{ Definition}
A continuum is a {\bf half-ray curve} if it is a continuous
1-1 image of the nonnegative reals.
\enddemo

% hereditarily atriodic

\demo{ Definition}
A continuum $X$ is said to be {\bf hereditarily atriodic} provided that each
nondegenerate subcontinuum of $X$ is atriodic.
\enddemo


% hereditarily decomposable

\demo{ Definition}
A continuum $X$ is said to be  {\bf hereditarily
decomposable} provided that each nondegenerate subcontinuum
of $X$ (as well as $X$ itself) is decomposable.
\enddemo

% hereditarily equivalent

\demo{ Definition}
A continuum is {\bf hereditarily equivalent} if it is
homeomorphic to each of its nondegenerate subcontinua.
\vertspace\enddemo


% hereditarily indecomposable

\demo{ Definition}
A continuum $X$ is said to be  {\bf hereditarily
indecomposable} provided that each
of its subcontinua is indecomposable.
\enddemo


% hereditarily unicoherent

\demo{ Definition}
A connected topological space is said to be
{\bf hereditarily unicoherent} provided that
each of its closed, connected subsets is unicoherent.
\enddemo

% HU-terminal

\demo{ Definition}
A subcontinuum $K$ of a hereditarily unicoherent continuum
$X$ is said to be a {\bf HU-terminal continuum}
of $X$ provided that $K$ is contained in an
irreducible subcontinuum of $X$ and for every
irreducible subcontinuum $I$ of $X$ containing $K$
there is a point $x\in X$ such that
$I$ is irreducible about the union $K\cup \{x\}$.
\enddemo



% homogeneous

\demo{ Definition}
A continuum $X$ is {\bf homogeneous} with respect to $\Cal F$ if,
for each two points $a$ and $b$ of $X$, there is a mapping $f$ in
$\Cal F$ of $X$ onto $X$ such that $f(a)=b$.
\enddemo



% indecomposable

\demo{ Definition}
A continuum $X$ is said to be  {\bf indecomposable} provided
that $X$ cannot be written as the union of two proper
subcontinua.
\enddemo



% irreducible

\demo{ Definition}
Let $X$ be a continuum and $A\subset X$. Then, $X$ is said
to be {\bf irreducible about} $A$ provided no proper
subcontinuum of $X$ contains $A$. A continuum $X$ is said to
be  {\bf irreducible} provided that $X$ is irreducinle about
$\{p,q\}$ for some $p,q \in X$.
\enddemo


% irreducible mapping

\demo{ Definition}
Let $X$ be a continuum.
A mapping $f: X \to f(X)=Y$ is said to be
{\bf irreducible} provided that for each
nondegenerate irreducible subcontinuum $I\subset Y$ the inverse image $f^{-1}(I)$
is irreducible.
\enddemo



% light

\demo{ Definition}
A mapping is {\bf light} if each point inverse is totally
disconnected.
\enddemo



% like

\demo{ Definition}
If $X$ is a metric space, a mapping $f$ from $X$ to a space $Y$
is an {\bf $\eps$-map} if, for each point $y$ of $Y$,
$\diam\(f^{-1}\(y\)\) \leq \eps$.
If $\Cal C$ is a collection of continua, a continuum $M$ is
{\bf $\Cal C$-like} if, for every positive number $\eps$,
there exists an $\eps$-map of $M$ onto an element of
$\Cal C$.  In particular, a continuum is {\bf tree-like} if, for
some collection $\Cal C$ of trees, $M$ is $\Cal C$-like.
\vertspace\enddemo





% locally confluent

\demo{ Definition}
A mapping $f$ from a space $X$ onto a space $Y$ is {\bf locally
confluent} if, for each point $y$ of $Y$, there is an open set
$U$ containing $y$ such that $f$ restricted to the set 
$f^{-1}(\, \Cl{U} \,)$ is confluent.
\enddemo

% local separating point

\demo{ Definition}
A point $p$ of a locally compact separable metric space $L$ is a
{\bf local separating point} of $L$ provided there exists an open
set $U$ of $L$ containing $p$ and two points $x$ and $y$ of the
component containing $p$ of $U$ such that $U \setminus \{ p \}$
is the sum of two mutually separated point sets, one containing
$x$ and the other containing $y$.
\enddemo

% mean
\demo{ Definition}
A {\bf mean} on topological space $X$ is a continuous symmetric function
$\mu:X\times X\to X$ which satisfies $\mu(x,x)=x$ for every $x\in X$.
\enddemo

% MO-mapping

\demo{ Definition}
An {\bf MO-mapping} is a composition $gf$ where $g$ is monotone
and $f$ is open.
\enddemo

% model

\demo{ Definition}
A continuum $M$ is said to be a {\bf model} for a class $\Cal{C}$ of continua
provided that for every $C\in\Cal{C}$ there exist surjective mapping $f:M\to
C$.
\enddemo

% monotone

\demo{ Definition}
A continuous function $f:X \to Y$ is said to be  {\bf
monotone} provided  $f^{-1}(y)$ is connected for each $y \in
Y$.
\enddemo

% mutually aposyndetic

\demo{ Definition}
A continuum $M$ is {\bf mutually aposyndetic} if, for each two
points $A$ and $B$ of $M$, there exist mutually exclusive
subcontinua $H$ and $K$ of $M$ containing $A$ and $B$,
respectively, in their interiors.  A continuum is {\bf strictly
non-mutually aposyndetic} if each two of its subcontinua with
interiors intersect.
\enddemo


% open

\demo{ Definition}
A mapping $f$ of a space $X$ into a space $Y$ is {\bf open}
if, for each open $A$ in $X$ the image $f(A)$ is open in
$Y$.
\enddemo


% order

\demo{ Definition}
Let $(X,T)$ be a topological space, and let $A \subset X$.
Let $\beta$ be a cardinal number.
We say that $A$  {\bf is of order less than equal to $\beta$ in} $X$,
written ord($A,X$)$\le \beta$, provided that for each $U \in T$ such that
$A\subset U$, there exists $V\in T$ such that
$A \subset V \subset U$ and $|Bd(V)|\le \beta$.

We say that $A$ {\bf is of order $\beta$ in} $X$, written
ord($A,X$)$= \beta$, provided that ord($A,X$)$\le \beta$ and
ord($A,X$)$\not \le \alpha$ for any cardinal number $\alpha < \beta$.
\enddemo


% pseudo-arc

\demo{ Definition}
A {\bf pseudo-arc} is the only (up to homeomorphism)
hereditarily indecomposable arc-like continuum.
\enddemo


% property $A$

\demo{ Definition}
Suppose $M$ is a connected metric space and $x$ and $y$ are two
points of $M$.  If, for each connected and closed subset of $K$
of $M-\{x\}$ containing $y$, $M$ is aposyndetic at $x$ with
respect to $K$, then $M$ is said to have {\bf property $A$} at
$x$ with respect to $y$.  A continuum has {\bf property $A$} if
it has property $A$ at each point with respect to every other
point.
\enddemo

% property $F$

\demo{ Definition}
A mapping $f$ of a continuum $X$ onto a continuum $Y$ has {\bf
property $F$} provided that there exists a point $x$ in $X$ such
that if $K$ is a subcontinuum of $Y$ containing $f(x)$ and $C$ is
the component of $f^{-1}(K)$ containing $x$, then $f(C)=K$.
\enddemo

% property of Kelley

\demo{ Definition}
A continuum $X$ is said to have {\bf property of Kelley} provided that for
every $a\in X$ and for every subcontinuum $A\subset X$ such that $a\in A$ and
for every sequence $a_n\in X$ which converges to $a$ there exist a sequence of
subcontinua $A_n\subset X$ such that $a_n\in A_n$ and $A_n$ converges to $A$
(in the Hausdorff metric).
\enddemo

% pseudo-confluent

\demo{ Definition}
A mapping from $X$ to $Y$ is {\bf pseudo-confluent} if every
irreducible continuum in $Y$ is the image of a continuum in $X$.
\enddemo



%  rational

\demo{ Definition}
A continuum is {\bf rational} if each two of its points can be
separated by a countable point set.
\enddemo

% real curve

\demo{ Definition}
A continuum is a {\bf real curve} if it is a continuous 1-1
image of the real line.
\enddemo



% (monotonely) refinable

\demo{ Definition}
A map $r$ from a compact metric space $X$ onto a compact metric
space $Y$ is {\bf (monotonely) refinable} if, for each positive
number $\eps$, there is a (monotone) $\eps$-map $f$ from
$X$ onto $Y$ such that, for each $x$ in $X$,
$d\( f\(x\),r\(x\) \) < \eps$.
\enddemo


% regular

\demo{ Definition}
A continuum is {\bf regular} if each two of its points can be
separated by a finite point set.
\enddemo



% scattered

\demo{ Definition}
A point set $X$ is {\bf scattered} if every subset $Y$ of $X$ has
a point that is not a limit point of $Y$.
\enddemo



% semi-aposyndetic

\demo{ Definition}
A space $X$ is {\bf semi-aposyndetic} if, for each two of its
points, $X$ is aposyndetic at one of them with respect to the
other.
\vertspace\enddemo



% semi-continuum

\demo{ Definition}
A {\bf semi-continuum} is a continuumwise connected point set.
\enddemo


% semi-locally connected

\demo{ Definition}
A continuum $M$ is {\bf semi-locally connected} at the point $p$
if, for every domain $D$ containing $p$, there exits a domain $E$
lying in $D$ and containing $p$ such that $M \setminus E$ has
only a finite number of components.
A continuum is {\bf totally non-semi-locally connected} if it is
not semi-locally connected at any point.
\enddemo

% \Sierpinski\  curve

\demo{ Definition}
The {\bf \Sierpinski\  curve} is a locally connected planar curve
with infinitely many complementary domains whose boundaries are
mutually exclusive simple closed curves.
\enddemo



% $\sigma$-connected

\demo{ Definition}
A  topological space $(X,\tau)$ is said to be {\bf
$\sigma$-connected} provided that $X$ is not the union of
more than one and at most countably many nonempty, mutually
disjoint, closed subsets.
\enddemo

% simple closed curve

\demo{ Definition}
A  {\bf simple closed curve} is any space which is homeomorphic to the
unit circle.
\enddemo



% simple triod

\demo{ Definition}
The union of three arcs emanating from a point v is called a
{\bf simple triod} provided the singleton {v} is the
intersection of any two of the arcs ; the point v is then
called the {\bf top} of the simple triod, and the arcs are
called its {\bf arms}.
\enddemo


% smooth (dendroid)

\demo{ Definition}
A dendroid $X$ is {\bf smooth} if there is a point $p$ in $X$
such that, if $x_1, x_2, \dots$ is a sequence of points of $X$
converging to the point $x$, then the superior limit of the arcs
$[p,x_n]$ is the arc $[p,x]$.
\enddemo

% strongly monotone

\demo{ Definition}
Let $X$ be a continuum.
A mapping $f: X \to f(X)=Y$ is said to be
{\bf strongly monotone} provided that for each
nondegenerate irreducible subcontinuum $I\subset Y$ the inverse image $f^{-1}(I)$
is irreducible.
\enddemo

% strongly unicoherent

\demo{ Definition}
A continuum $X$ is {\bf strongly unicoherent} provided $X$ is
unicoherent and each proper subcontinuum with interior is
unicoherent.
\enddemo


% Suslinean

\demo{ Definition}
A continuum is {\bf Suslinean} if it does not contain uncountably
many mutually exclusive nondegenerate subcontinua.
\enddemo


% terminal continuum

\demo{ Definition}
A proper subcontinuum $K$ of a continuum $X$ is
said to be a {\bf terminal continuum} of $X$ provided
that if whenever $A$ and $B$ are proper subcontinua
of $X$ having union equal to $X$ such that
$A\cap K \ne \emptyset \ne B \cap K$, then either
$X=A\cup K$ or $X=B\cup K$.
\enddemo


% thin

\demo{ Definition}
A topological space $S$ is {\bf thin} if and only if for each two
homeomorphic subsets $A$ and $B$ of $S$, there is a homeomorphism
$h$ of $S$ onto itself such that $h(A)=B$.
\enddemo



% tree

\demo{ Definition}
A  {\bf tree} is a graph which contains no simple closed
curve.
\enddemo


% triod

\demo{ Definition}
A continuum is called a {\bf triod} provided it contains a
subcontinuum whose complement is the union of three nonempty
pairwise disjoint open sets.
\enddemo


% unicoherent

\demo{ Definition}
A connected topological space $S$ is said to be
{\bf unicoherent} provided that whenever $A$ and $B$
are closed, connected subsets of $S$ such that
$S=A\cup B$, then $A\cap B$ is connected.
\enddemo



% weak cut point

\demo{ Definition}
A point $x$ of a continuum $M$ is a {\bf weak cut point} of $M$
if there are two points $p$ and $q$ of $M$ such that every
subcontinuum of $M$ that contains both $p$ and $q$ also contains
$x$.
\enddemo

% weakly chainable

\demo{ Definition}
A continuum is {\bf weakly chainable} if it is a continuous image
of a chainable continuum.
\enddemo



% weakly Suslinian

\demo{ Definition}
A continuum $X$ is said to
be {\bf weakly Suslinian} provided that $X$ is not
the union of more than one pairwise disjoint
nondegenerate subcontinua of $X$.
\enddemo


% widely connected

\demo{ Definition}
A nondegenerate connected space $X$ is {\bf widely
connected} if each nondegenerate connected subset of $X$ is
dense in $X$.
\enddemo

%  width $w(X)$

\demo{ Definition}
For any compact metric space $X$, the {\bf width $w(X)$} of $X$
is the l.u.b. of the set of all real numbers $a$ which satisfy
the following condition: for each $\eps > 0$, there exists a
finite open cover $\Cal C$ of $X$ such that
$\mesh(\Cal C) < \eps$ and for each chain $\Cal C'$ which is
a subcollection of $\Cal C$ there is a member $A$ of $\Cal C$
such that $d(A,{\Cal C'}^*) \geq a$.
\enddemo



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