Closure of Image under Continuous Mapping is not necessarily Image of Closure
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Theorem
Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $H \subseteq S_1$ be a subset of $S_1$.
Let $\map \cl H$ denote the closure of $H$.
Let $f: T_1 \to T_2$ be a continuous mapping.
Then it is not necessarily the case that:
- $f \sqbrk {\map \cl H} = \map \cl {f \sqbrk H}$
Proof
Let $\R$ be the real numbers under the usual (Euclidean) topology.
Let $f: \R \to \R$ be the (real) hyperbolic tangent function:
- $\forall x \in \R: \map f x = \tanh x$
It is accepted that $f$ is continuous.
Let $H \subseteq \R$ be the subset of $\R$ defined as:
\(\ds H\) | \(=\) | \(\ds \openint 0 \to\) | which is an open set in $\R$. | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \cl H\) | \(=\) | \(\ds \hointr 0 \to\) | Definition of Closure (Topology) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds f \sqbrk H\) | \(=\) | \(\ds \openint 0 1\) | Definition of Real Hyperbolic Tangent | ||||||||||
\(\ds f \sqbrk {\map \cl H}\) | \(=\) | \(\ds \hointr 0 1\) | Definition of Real Hyperbolic Tangent | |||||||||||
\(\ds \map \cl {f \sqbrk H}\) | \(=\) | \(\ds \closedint 0 1\) | Definition of Closure (Topology) |
As can be seen:
- $f \sqbrk {\map \cl H} \ne \map \cl {f \sqbrk H}$
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: Exercise $3.9: 27 \ \text {(i)}$