L-convergence

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Let $(X,\tau)$ be a topological space, let $\mathcal{A}=\{A_i\}_{i=1}^{\infty}$ be a sequence of subsets of $X$, and let $A \subset X$. We say that $\mathcal{A}$ is $L$-convergent in $X$ to $A$ (denoted by $\lim A_i=A$) provided that $$\liminf \mathcal{A} = A = \limsup \mathcal{A},$$ where $\liminf$ and $\limsup$ denote the limit inferior and limit superior respectively.


The relationship between $L$-convergence and $2^X$-convergence (where $L$-convergence takes place in $X$ and $2^X$-convergence takes place in $CL(X)$)

Theorem:Let $(X,\tau)$ be a regular topological space . Then $2^X$-convergence in $CL(X)$ implies $L$-convergence in $X$.