Weaker and Stronger Topologies

Weaker and Stronger Topologies

Definition: Let $X$ be a topological space and let $\tau_1$ and $\tau_2$ be topologies on $X$. Then $\tau_1$ is said to be Weaker or Coarser than $\tau_2$ if $\tau_1 \subseteq \tau_2$, and $\tau_1$ is said to be Stronger or Finer than $\tau_2$ if $\tau_1 \supseteq \tau_2$.
Proposition 1: Let $X$ be a topological space and let $\tau_1$ and $\tau_2$ be topologies on $X$. Suppose that $\tau_1$ is weaker than $\tau_2$. Then if $f : X \to \mathbb{C}$ is continuous with respect to the topology $\tau_1$ then $f$ is continuous with respect to the topology $\tau_2$.
  • Proof: Let $f : X \to \mathbb{C}$ and let $U \subseteq \mathbb{C}$ be open. Since $f$ is continuous with respect to the topology $\tau_1$ we have that $f^{-1}(U) \in \tau_1$. But then since $\tau_1$ is weaker than $\tau_2$ we have that
(1)
\begin{align} \quad f^{-1}(U) \in \tau_1 \subseteq \tau_2 \end{align}
  • So $f$ is continuous with respect to the topology $\tau_2$.
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