# How to prove two topologies $\mathcal{T}_1,\mathcal{T}_2$ are not equal

Let $C[0; 1]$ be the set of all continuous real-valued functions on $[0; 1]$.

(i) Show that the collection $M$, where $M = \{M(f,\varepsilon ) : \text{$f\in C\left[0; 1\right ]$and$\varepsilon $is a positive real number}\}$ and $M(f,\varepsilon) =\{g : \text{$g\in C\left[0; 1\right ]$and$\int_{0}^{1}\left|f-g\right| < \varepsilon $}\}$, is a basis for a topology $\mathcal{T}_{1}$ on $C[0; 1]$.

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