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Example: Example of Continuous Functions in Topological Spaces

(related to Chapter: Continuity)

Example 1

The identity function $id:X\to X$ on any given topological space $(X,\mathcal O)$ is continuous.

Example 2

If $(X,\mathcal O_X)$ and $(Y,\mathcal O_Y)$ are topological spaces, then the constant function $f:X\to Y$, $f(x)=y\in Y$ for all $x\in X$ is continuous, since for an $B\subseteq Y$ we have * either $f^{-1}[B]=\emptyset,$ if $y\not\in B,$ * or $f^{-1}[B]=X,$ if $y\in B.$

Both, $\emptyset$ and $X$ are open in $X$ by definition of topological space.

Example 3

Consider $(X,\mathcal O_X)$ and $(Y,\mathcal O_Y)$ as topological spaces. The continuity of $f:X\to Y$ depends on both topologies, $\mathcal O_X$ and $\mathcal O_Y.$ For instance:

• If $\mathcal O_X$ is the discrete topology and $\mathcal O_Y$ is the indiscrete topology, then every function $f:X\to Y$ is continuous.
• If $\mathcal O_X$ is the indiscrete topology and $\mathcal O_Y$ is the discrete topology, then only the constant functions $f:X\to Y$ are continuous.

Example 4

If $f:X\to Y$ is a continuous function defined on two topological spaces $(X,\mathcal O_X)$ and $(Y,\mathcal O_Y)$, then $f$ remains continuous if we replace the topology $\mathcal O_X$ by a finer one, and the topology $\mathcal O_Y$ by a coarser one.

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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
3. Grotemeyer, K.P.: "Topologie", B.I.-Wissenschaftsverlag, 1969