239
Transfinite Induction and General Topology
(3) We call X a T 2 -space or a Hausdorff space if, whenever x and y are
distinct points of X, there are disjoint neighbourhoods of x and y.
One of the important properties of Hausdorff spaces is that stated in the
following result.
A.4.2 Theorem A topological space is Hausdorff if and only if every convergent net in X has a unique limit.
On the other hand, it is important that there not be too many open sets
in a certain sense.
A.4.3 Definition Let X be a topological space. Then an open cover {U i |
i ∈ I} of X is a collection of open sets U i such that
U i = X. A subcover
i∈I
of an open cover {U i | i ∈ I} is a cover {V j | j ∈ J }, where J ⊆ I. We call a
topological space X compact if every open cover of X has a finite subcover.
A.5 Subspaces and Products
There are several ways in which one can create new topological spaces
from given ones. We discuss here just two of these, namely, the process of
forming subspaces of topological spaces and the process of forming products
of families of topological spaces.
A.5.1 Definition Let (X, τ ) be a topological space, and let S ⊆ X be a
subset of X. Then the collection τ S = {S ∩ O | O ∈ τ } gives a topology on S,
called the relative topology or the subspace topology for S. The space (S, τ S )
is called a subspace of (X, τ ) or just a subspace of X.
Whenever one has a topological space X and a subset S of X, it will be
assumed that S has been endowed with the subspace topology of X unless
stated to the contrary. Notice that the sets S ∩ O, where O is open in X, need
not be open in X unless S itself is an open set of X.
Now suppose that X i is a topological space for each i, where i is an element
of some index set I. As usual, we denote the product of the family {X i | i ∈ I}
s
of sets by
X i = {f : I →
X i | f (i) ∈ X i }. Associated with any such
i∈I
i∈I
s
product are the mappings π j , j ∈ I, where π j :
X i → X j is defined by
i∈I
π j (f ) = f (j). Indeed, π j is termed the projection on the j-th factor .
s
There is a natural topology one can define on
X i determined by the
i∈I
projections as follows. Choose any finite set {i 1 , . . . , i n } of elements of I, and
choose corresponding open sets U ij in X ij , for j = 1, . . . , n. Then we take
the collection of sets of the form π
−1
) ∩ . . . ∩ π
−1
) as a base for
i1 (U i1
in (U in
s
a topology on
called the product topology or the Tychonoff product
i∈I X i
Transfinite Induction and General Topology
(3) We call X a T 2 -space or a Hausdorff space if, whenever x and y are
distinct points of X, there are disjoint neighbourhoods of x and y.
One of the important properties of Hausdorff spaces is that stated in the
following result.
A.4.2 Theorem A topological space is Hausdorff if and only if every convergent net in X has a unique limit.
On the other hand, it is important that there not be too many open sets
in a certain sense.
A.4.3 Definition Let X be a topological space. Then an open cover {U i |
i ∈ I} of X is a collection of open sets U i such that
U i = X. A subcover
i∈I
of an open cover {U i | i ∈ I} is a cover {V j | j ∈ J }, where J ⊆ I. We call a
topological space X compact if every open cover of X has a finite subcover.
A.5 Subspaces and Products
There are several ways in which one can create new topological spaces
from given ones. We discuss here just two of these, namely, the process of
forming subspaces of topological spaces and the process of forming products
of families of topological spaces.
A.5.1 Definition Let (X, τ ) be a topological space, and let S ⊆ X be a
subset of X. Then the collection τ S = {S ∩ O | O ∈ τ } gives a topology on S,
called the relative topology or the subspace topology for S. The space (S, τ S )
is called a subspace of (X, τ ) or just a subspace of X.
Whenever one has a topological space X and a subset S of X, it will be
assumed that S has been endowed with the subspace topology of X unless
stated to the contrary. Notice that the sets S ∩ O, where O is open in X, need
not be open in X unless S itself is an open set of X.
Now suppose that X i is a topological space for each i, where i is an element
of some index set I. As usual, we denote the product of the family {X i | i ∈ I}
s
of sets by
X i = {f : I →
X i | f (i) ∈ X i }. Associated with any such
i∈I
i∈I
s
product are the mappings π j , j ∈ I, where π j :
X i → X j is defined by
i∈I
π j (f ) = f (j). Indeed, π j is termed the projection on the j-th factor .
s
There is a natural topology one can define on
X i determined by the
i∈I
projections as follows. Choose any finite set {i 1 , . . . , i n } of elements of I, and
choose corresponding open sets U ij in X ij , for j = 1, . . . , n. Then we take
the collection of sets of the form π
−1
) ∩ . . . ∩ π
−1
) as a base for
i1 (U i1
in (U in
s
a topology on
called the product topology or the Tychonoff product
i∈I X i