Top (Rev #17, changes) in nLab

Context

Topology

• Definition
• Properties
  • Universal constructions
  • Relation with SetSet
• Related concepts
• References

Poposition

Let X •:I⟶TopX_\bullet \colon I \longrightarrow Top be a diagram in Top, with components denoted X i=(S i,τ i)X_i = (S_i, \tau_i), where S i∈SetS_i \in Set and τ i\tau_i a topology on S iS_i. Then:

1. The limit of X •X_\bullet exists and is given by the topological space whose underlying set is the limit in Set of the underlying sets in the diagram, and whose topology is the initial topology for the functions p ip_i which are the limiting cone components:

   lim⟵ i∈IS i p i↙ ↘ p j S i ⟶ S j. \array{ && \underset{\longleftarrow}{\lim}_{i \in I} S_i \\ & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}} \\ S_i && \underset{}{\longrightarrow} && S_j } \,.

   Hence

   lim⟵ i∈IX i≃(lim⟵ i∈IS i,τ initial({p i} i∈I)) \underset{\longleftarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)

2. The colimit of X •X_\bullet exists and is the topological space whose underlying set is the colimit in Set of the underlying diagram of sets, and whose topology is the final topology τ final\tau_{final} for the component maps ι i\iota_i of the colimiting cocone

   S i ⟶ S j ι i↘ ↙ ι j lim⟶ i∈IS i. \array{ S_i && \longrightarrow && S_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && \underset{\longrightarrow}{\lim}_{i \in I} S_i } \,.

   Hence

   lim⟵ i∈IS i≃(lim⟵ i∈IS i,τ final({ι i} i∈I)) \underset{\longleftarrow}{\lim}_{i \in I} S_i \simeq \left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{final}(\{\iota_i\}_{i \in I})\right)

Example

• The limit over the empty diagram in TopTop is the point *\ast with its unique topology.

• For S∈SetS \in Set, the SS-index coproduct of the point is the set SS itself equipped with the initial topology, hence is the discrete topological space on SS.

Definition

Write

U:Top⟶Set U \colon Top \longrightarrow Set

for the forgetful functor that sends a topological space X=(S,τ)X = (S,\tau) to its underlying set U(X)=SU(X) = S.