Spazio di Hausdorff

In topologia, uno spazio di Hausdorff, detto anche spazio separato e spesso abbreviato con T₂, è uno spazio topologico nel quale per due punti distinti si possono sempre trovare degli intorni aperti disgiunti. Il nome è in onore del matematico tedesco Felix Hausdorff, 1868-1942.

La maggior parte degli spazi considerati in analisi matematica è di Hausdorff, tanto che Felix Hausdorff incluse l'assioma di separazione nella sua definizione originaria di spazio topologico (1914). Successivamente però si è mostrato utile considerare anche spazi non separati.

Inoltre, il prodotto di ogni famiglia di spazi di Hausdorff è uno spazio di Hausdorff.^[1]

Definizione

Uno spazio di Hausdorff è uno spazio topologico che soddisfa il seguente assioma di separazione:

\forall \,x,y\in X\,|\,x\neq y\,\exists \,U,V{\text{ intorni aperti di }}x,y\,|\,U\cap V=\varnothing

.^[2]^[3]

Uno spazio di Hausdorff è anche uno spazio T1, infatti basta dimostrare che i punti sono chiusi: ma questo è vero poiché esistono degli intorni disgiunti del punto in questione e di qualunque punto dell'insieme complementare e dunque il complementare è intorno di ogni suo punto, allora è aperto e il singolo punto è un chiuso.

Esempi

I numeri reali, con la ordinaria topologia in cui gli insiemi aperti sono esattamente tutte le unioni arbitrarie di intervalli aperti, sono uno spazio di Hausdorff: Dati due numeri reali distinti x e y, x≠y, sia d = |x - y| / 2 la metà della loro distanza; allora gli intervalli U = ]x - d, x + d[ e V = ]y - d, y + d[ sono intorni disgiunti di x e y.

Un ragionamento simile mostra che ogni spazio metrico, quindi in particolare anche ogni spazio euclideo, è uno spazio di Hausdorff: dati due punti, si considerano le sfere aperte attorno a questi punti con raggio uguale alla metà della loro distanza; la disuguaglianza triangolare assicura che le due sfere sono disgiunte.

Non tutti gli spazi topologici sono di Hausdorff: un controesempio semplice è dato da uno spazio di almeno due punti X dotato della topologia banale {∅, X}. Un controesempio più interessante è la topologia di Zariski in geometria algebrica.

Note

^ Topological Spaces, pagina 281, 17.A..
^ W. Rudin, Pag. 36.
^ E. Sernesi, Pag. 88.

Bibliografia

Walter Rudin, Real and Complex Analysis, Mladinska Knjiga, McGraw-Hill, 1970, ISBN 0-07-054234-1.

Voci correlate

Collegamenti esterni

Hausdorff, spazio di, in Enciclopedia della Matematica, Istituto dell'Enciclopedia Italiana, 2013.
(EN) Stephan C. Carlson, Hausdorff space, su Enciclopedia Britannica, Encyclopædia Britannica, Inc.
(EN) Eric W. Weisstein, Spazio di Hausdorff, su MathWorld, Wolfram Research.

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