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Definition: Ordering of Topologies

For any two topologies $\mathcal O_1$ and $\mathcal O_2$ defined on a set $X$, we can define a partial order using the subset relation:

$$\mathcal O_1\le \mathcal O_2\Longleftrightarrow \mathcal O_1\subseteq \mathcal O_2.$$

If $\mathcal O_1\le \mathcal O_2$, then we say

• $\mathcal O_1$ is smaller (or weaker or coarser) and
• $\mathcal O_2$ is larger (or stronger or finer).

Notes:

• The ordering $\le$ is only a partial order, since two topologies might be not comparable (i.e. if we choose the sets $\mathcal O_1$ and $\mathcal O_2$ in such a way that neither $\mathcal O_1\subseteq \mathcal O_2$, nor vice versa).
• For every set $X$, the largest topology (finest) topology is the discrete topology, the smallest (coarsest) the indiscrete topology.

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References

Bibliography

1. Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970