Teilraumtopologie

Im mathematischen Teilgebiet der Topologie versteht man unter der Teilraumtopologie (auch induzierten Topologie, relativen Topologie, Spurtopologie oder Unterraumtopologie) die natürliche Struktur, die eine Teilmenge eines topologischen Raumes „erbt“. Die Teilraumtopologie ist eine spezielle Initialtopologie.

Formale Definition

Es sei $X$ die Grundmenge eines topologischen Raums $\left(X,{\mathcal {T}}\right)$ und $Y\subseteq X$ eine Teilmenge. Dann ist die Teilraumtopologie auf $Y$ die Topologie

{\mathcal {T}}_{Y}=\{O\cap Y\mid O\in {\mathcal {T}}\}.

Die offenen Teilmengen von $Y$ sind also genau die Schnitte der offenen Teilmengen $O$ von $X$ mit $Y$ .

Eigenschaften

Die Teilraumtopologie auf einer Teilmenge $Y\subseteq X$ eines topologischen Raumes $X$ ist die gröbste Topologie, für die die Inklusionsabbildung

Y\to X,\quad y\mapsto y

stetig ist.

Ist $Y$ eine offene Teilmenge eines topologischen Raumes $X$ , so ist eine Teilmenge $U\subseteq Y$ genau dann offen in der Teilraumtopologie von $Y$ , wenn $U$ als Teilmenge von $X$ offen ist.
Ist $Y$ eine abgeschlossene Teilmenge eines topologischen Raumes $X$ , so ist eine Teilmenge $Z\subseteq Y$ genau dann abgeschlossen in der Teilraumtopologie von $Y$ , wenn $Z$ als Teilmenge von $X$ abgeschlossen ist.

Beispiele

Man stelle sich ein Blatt Papier ohne Rand als zweidimensionales Objekt vor. Im $\mathbb {R} ^{3}$ ist dies keine offene Menge. Betrachtet man aber die Topologie bezüglich der Ebene, in der sich das Blatt befindet, so liegt eine offene Menge vor.
Die Teilraumtopologie auf $\mathbb {Z} \subset \mathbb {R}$ ist die diskrete Topologie, d. h. alle Teilmengen von $\mathbb {Z}$ sind offen als Teilmengen des topologischen Raumes $\mathbb {Z}$ . Beispielsweise ist die Menge $\{0\}$ eine offene Teilmenge von $\mathbb {Z}$ , weil sie Schnitt der offenen Teilmenge $\left(-{\tfrac {1}{2}},{\tfrac {1}{2}}\right)$ von $\mathbb {R}$ mit $\mathbb {Z}$ ist.

Literatur

Boto von Querenburg: Mengentheoretische Topologie (= Springer-Lehrbuch). 3., neu bearbeitete und erweiterte Auflage. Springer, Berlin u. a. 2001, ISBN 3-540-67790-9.

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