----- > [!proposition] Proposition. ([[sequence convergence is invariant under topological coarsening, but not under finening]]) > Let $(a_{n})_{n \in \mathbb{N}}$ be a [[sequence]] of points in a [[topological space]] $(X, \tau)$ converging to $a_{\infty} \in X$. > 1. If we replace the topology $\tau$ on $X$ with a [[comparable topologies|coarser topology]] $\tau'$ then the sequence $(a_{n})_{n \in \mathbb{N}}$ will still [[converge]] to $a_{\infty}$. > 2. However, if we replace the topology on $X$ with any [[comparable topologies|finer]] topology, then, then the sequence $(a_{n})_{n \in \mathbb{N}}$ may no longer [[converge]] to $a_{\infty}$. > [!proof]- Proof. ([[sequence convergence is invariant under topological coarsening, but not under finening]]) > > >1. xLet $\tau' \subset \tau$ be the coarsened topology. Let $U \in \tau'$ be an open [[neighborhood]] of $a_{\infty}$ wrt $\tau'$. Since $U \in \tau$, we can find $N \in \mathbb{N}$ large enough that for all $n>N$ we have $a_{n} \in U$. But this means exactly that $(a_{n}) \to a_{\infty}$. >2. Counterxample: $\mathbb{R}$, the sequence $\left( \frac{1}{n}: n \in \mathbb{N} \right)$ converges to $0$ wrt the standard topology. But recall that any sequence converges wrt the [[discrete topology]] iff it is eventually constant, which this sequence is not. So it does not converge wrt the discrete topology (which is obviously finer than the standard topology). ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```