# Random math: 25

Source of most content: Gunhee Cho.

# Product Topology

Def. Given an index set $J$, let $(A _\alpha) _{\alpha \in J}$ be a seq. of sets. Then, Generalized Cartesian product of $(A _\alpha) _{\alpha \in J}$ is $\Pi _{\alpha \in J} A _\alpha := \{ f: J \rightarrow \cup _{\alpha \in J} A _\alpha : \forall \alpha, f(\alpha) \in A _\alpha \}$. For each $j \in J$, the function $\pi_j : \Pi _{\alpha \in J} A _\alpha \rightarrow X_j$ defined by $\pi_j(f) = f(j)$ is called $j$-th projection map.

Recall that given T.S. $(X, T_X)$ and $(Y, T_Y)$, we define $T_{box}$ on $X\times Y$ generated by $\mathcal{B}= \{ U \times V : U \in T_X, V \in T_Y \}$ Consider $\mathcal{B}$ is generalized by subbasis $\mathcal{S} := \{ \pi_1^{-1}(U)= U \times Y, \pi_2^{-1}(V)=X\times V: U \in T_X, V \in T_Y \}$.

Def. Given an index set $J$, let $(X_\alpha, T_\alpha)$ be T.S. for all $\alpha \in J$.

1. (Box Topology) $(\Pi _{\alpha \in J} X _\alpha, T _{box} )$ where $T _{box}$ is generated by basis $\mathcal{B}:=\{\Pi _{\alpha \in J} U _\alpha : U _\alpha \in T _\alpha \}$.
2. (Product Topology) $(\Pi _{\alpha \in J} X _\alpha, T _{prod} )$ where $T _{prod}$ is generated by subbasis $\mathcal{S}:=\{\pi^{-1} _{\alpha \in J} (U _\alpha) : U _\alpha \in T _\alpha \}$.

From the definition, $T_{prod} = \{ \Pi _{\alpha \in J} U _\alpha : U _\alpha \in T _X, U _\alpha = X _\alpha \text{ all but finitely many } \alpha \}$.

NB1; By definition, $T_{prod} \subsetneq T_{box}$ for infinite product of $X_\alpha$.
NB2; For $X_1, \ldots, X_n$, $T_{prod} = T_{box}$.

## Proposition

$\text{If }X_\alpha \text{ is Hausdorff }\forall \alpha, \text{Then, }\Pi X_\alpha \text{ is Hausdorff in both box and product topologies}. \nonumber$

### Proof

Take two distinct points $x, y \in \Pi X_\alpha$. Then, $\exists j \in J$ s.t. $x_{j} \ne y_{j}$. Since $X_j$ is Hausdorff, $\exists$ nbhd $U_j$ and $V_j$ of $x_j$, $y_j$ s.t. $x_j \in U_j, y_j \in V_j$ and $U_j \cap V_j = \emptyset$. Let $U_\alpha = V_\alpha = X_\alpha$ for $\alpha \in J \setminus \{ j \}$ and $U = \Pi_{\alpha \in J} U_\alpha$, $V = \Pi_{\alpha \in J} V_\alpha$. Then, $U$ and $V$ are open sets in $\Pi X_\alpha$ and $x \in U$, $y \in V$, $U \cap V = \emptyset$. Thus, $\Pi X_\alpha$ is Hausdorff.

# Initial Topology

Def. Given a function $f: X\rightarrow Y$ on $(X,T_X), (Y, T_Y)$. Initial Topology $T_X$ is a coarsest topology for which $f$ is continous.

NB3; We can guess $T_X = \{ f^{-1}(V) : V \in T_Y \}$ would be initial topology.
NB4; Consider an inclusion map $j : A \rightarrow (X, T_X) (x \mapsto x)$. Then, subspace topology $T_A := \{ j^{-1}(U) : U \in T_X\}$ $= \{ U \cap A : U \in T_X\}$, i.e., subspace topology is an initial topology which makes inclusion map continous.
NB5; Similarly, product topology is an initial map which makes projection map continous.

# Final Topology

Def. Given a function $f: X\rightarrow Y$ on $(X,T_X), (Y, T_Y)$. Final Topology $T_Y$ is a finest topology for which $f$ is continous.

NB6; We can guess $T_Y = \{ V \subseteq Y : f^{-1}(V) \subseteq T_X \}$ would be final topology.