# Definition: Open, Closed, Clopen

Let $(X,\mathcal{O})$ be a topological space. Every element of the topology $\mathcal O$ is called open. A subset $U\subseteq X$ is called closed, if its set complement $U^C$ is open, i.e. an element of the topology $\mathcal O.$

A subset of $X$, which is both, open and closed, is called clopen.

### Notes

• By definition of topology, it is clear that clopen sets exist. Depending on the specific collection of subsets of $X$ in the topology $\mathcal O$, there might be subsets of $X$ that are both, closed (since their complement is open) and open (since they are themselves elements of $\mathcal O$).
• Moreover, there might be subsets of $X$ which are neither open (since they are not elements of the topology $\mathcal O$), nor closed (since their complement is neither an element of this topology).

Axioms: 1
Definitions: 2 3 4 5 6 7 8 9
Examples: 10
Parts: 11
Proofs: 12 13 14 15 16 17 18 19 20 21
Propositions: 22 23 24 25 26 27

Github: ### References

#### Bibliography

1. Forster Otto: "Analysis 2, Differentialrechnung im $$\mathbb R^n$$, Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984
2. Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
3. Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition