Download 2 Continuous maps. 3 Induced topologies.

2 Continuous maps.
Definition 2.1 Let (X, U) and (Y, V) a function f : X → Y is called continuous if the
inverse image of any open set in Y is an open set in X. As a formula the definition is.
(1)
f
is continous ⇔ ∀(V ∈ V) f −1 (V ) ∈ U
Theorem 2.2 Let (X, U) and (Y, V) be topological spaces, and f : X → Y a function. The
following are equivalent.
(i)
f is continous
∀(x ∈ X) ∀(V ∈ N (f (x))) ∃(U ∈ N (x)) f (U ) ⊆ V
(ii)
∀(A ⊆ X)f (A) ⊆ f (A)
(iii)
3 Induced topologies.
Definition 3.1a (The direct image topology)
If f : X → Y is a function and U is a topology on X, the topology on Y induced by f is
the topology whose open sets are exactly the sets V ⊆ Y , with the property that their inverse
image are open sets in X. This topology is denoted f∗ (U). The formula below defines the direct
image topology.
V ∈ f∗ (U) ⇔ f −1 (V ) ∈ U
(1a)
Definition 3.1b (The inverse image or pull-back topology)
If f : X → Y is a function and V is a topology on Y , the topology on X induced by f is the
topology whose open sets are exactly the sets of the form f −1 (V ) for V ∈ V. This topology is
denoted f ∗ (U). The formula below defines the pull-back topology.
U ∈ f ∗ (V) ⇔ ∃(V ∈ V) U = f −1 (V )
(1b)
The following are facts about induced topologies. The proofs are left to the reader.
Theorem 3.2 Let (X, U) and (Y, V) be topological spaces, and f : X → Y a function. The
following are equivalent.
(i)
f : (X, U) → (Y, V)
(ii)
f∗ (U) ⊇ V
(iii)
is continous
U ⊇ f ∗ (V)
Corollary 3.3a If f : X → Y is a function and U is a topology on X, then the topology on
Y induced by f is the finest topology rendering f continuous. In other words.
(3a)
f∗ (U) = sup{W | f : (X, U) → (Y, W)
is continous}
Corollary 3.3b If f : X → Y is a function and V is a topology on Y , then the topology on
X induced by f is the coarsest topology rendering f continuous. In other words.
(3b)
f ∗ (V) = inf{W | f : (X, W) → (Y, V)
1
is continous}