◀ ▲ ▶Branches / Topology / Proposition: Equivalent Notions of Continuous Functions

Proposition: Equivalent Notions of Continuous Functions

Let $(X,\mathcal O_X)$ and $(Y,\mathcal O_Y)$ be topological spaces. The following definitions of a continuous function $f:X\to Y$ are equivalent:

1. The inverse image $f^{-1}[B]$ of every open set $B$ in $Y$ is open in $X.$
2. The inverse image $f^{-1}[B]$ of every closed set $B$ in $Y$ is closed in $X.$
3. The image of the closure of a subset $A\subset X$ is contained in the closure of the image of this subset, formally $f[A^-]\subseteq f[A]^-.$
4. For each $x\in X$ and each neighborhood $N_{f(x)}$ of $f(x)$ there exists a neighborhood $N_x$ of $x$ such that $f[N_x]\subseteq N_{f(x)}.$

Notes

• If the 4th condition is fulfilled for a particular point $x,$ $f$ is said to be continuous at the point $x.$

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1

Thank you to the contributors under CC BY-SA 4.0!

Github:

References

Bibliography

1. Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
2. Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition