(e) Connectedness
In the precedent several paragraphs we studied various building blocks and their
properties of the topological space and sets contained in the space. In this paragraph,
we briefly mention the relationship between the sets. The connectedness is an
important concept in the topology. The concept of the connectedness is associated
with the subspaces defined earlier in this section. In terms of the connectedness, a
topological space can be categorized into two main types: a subspace of the one type
is connected and that of the other is disconnected.
Let A be a subspace of T. Intuitively, for A to be connected is defined as follows:
Let z 1 and z 2 be any two points of A. If z 1 and z 2 can be joined by a continuous line
that is contained entirely within A (see Fig. 6.7a), then A is said to be connected. Of
the connected subspaces, suppose that the subspace A has the property that all the
points inside any closed path (i.e., continuous closed line) C within A contain only
the points that belong to A (Fig. 6.7b). Then, that subspace A is said to be simply
connected. A subspace that is not simply connected but connected is said to be
multiply connected. A typical example for the latter is a torus. If a subspace is given
as a union of two (or more) disjoint non-empty (open) sets, that subspace is said to be
disconnected. Notice that if two (or more) sets have no element in common, those
sets are said to be disjoint. Figure 6.7c shows an example for which a disconnected
subspace A is a union of two disjoint sets B and C.
Interesting examples can be seen in Chap. 20 in relation to the connectedness.
(a)
(b)
= ∪
(c)
Fig. 6.7 Connectedness of subspaces. (a) Connected subspace A. Any two points z 1 and z 2 of A can
be connected by a continuous line that is contained entirely within A. (b) Simply connected
subspace A. All the points inside any closed path C within A contain only the points that belong
to A. (c) Disconnected subspace A that comprises two disjoint sets B and C
194
6 Theory of Analytic Functions
In the precedent several paragraphs we studied various building blocks and their
properties of the topological space and sets contained in the space. In this paragraph,
we briefly mention the relationship between the sets. The connectedness is an
important concept in the topology. The concept of the connectedness is associated
with the subspaces defined earlier in this section. In terms of the connectedness, a
topological space can be categorized into two main types: a subspace of the one type
is connected and that of the other is disconnected.
Let A be a subspace of T. Intuitively, for A to be connected is defined as follows:
Let z 1 and z 2 be any two points of A. If z 1 and z 2 can be joined by a continuous line
that is contained entirely within A (see Fig. 6.7a), then A is said to be connected. Of
the connected subspaces, suppose that the subspace A has the property that all the
points inside any closed path (i.e., continuous closed line) C within A contain only
the points that belong to A (Fig. 6.7b). Then, that subspace A is said to be simply
connected. A subspace that is not simply connected but connected is said to be
multiply connected. A typical example for the latter is a torus. If a subspace is given
as a union of two (or more) disjoint non-empty (open) sets, that subspace is said to be
disconnected. Notice that if two (or more) sets have no element in common, those
sets are said to be disjoint. Figure 6.7c shows an example for which a disconnected
subspace A is a union of two disjoint sets B and C.
Interesting examples can be seen in Chap. 20 in relation to the connectedness.
(a)
(b)
= ∪
(c)
Fig. 6.7 Connectedness of subspaces. (a) Connected subspace A. Any two points z 1 and z 2 of A can
be connected by a continuous line that is contained entirely within A. (b) Simply connected
subspace A. All the points inside any closed path C within A contain only the points that belong
to A. (c) Disconnected subspace A that comprises two disjoint sets B and C
194
6 Theory of Analytic Functions