9c6c8fc5bd-p-08.pdf

# 9c6c8fc5bd-p-08.pdf - 1.1 GENERALIDADES 3 CAPITULO 1...

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1.1. GENERALIDADES 3 (3) Dado X = R definimos la topolog´ ıa usual τ de R como la colec- ci´on de todos los conjuntos que son intervalos abiertos o uniones arbitrarias de ellos. (4) Dado X = R 2 definimos la topolog´ ıa usual de R 2 como la colec- ci´on de todos los rect´angulos sin borde ] a, b [ × ] a , b [ o uniones arbitrarias de ellos. Ello es claramente generalizable a R n , n N , cuyos abiertos para la topolog´ ıa usual se definen como uniones arbitrarias de n - rect´angulos, cada uno de ´ estos definido como el producto cartesiano de n intervalos abiertos. Salvo especificaci´on contraria, R n se considerar´a dotado siempre de la topolog´ ıa usual. Como ejemplo veamos que (3) es una topolog´ ıa. Obviamente, se verif- ican los axiomas ( i ) ( =] a, a [, R =] - ∞ , [) y ( iii ). Por tanto, s´olo resta comprobar ( ii ). Para ello es suficiente demostrar que si U, V son abiertos y x U V entonces existe un intervalo abierto I x U V tal que x I x (pues en este caso U V = x U V I x τ ). Como
• Winter '15
• Punto, Conjunto, Conjunto finito, Intervalo, intervalo abierto ix, ∪x∈u ∩v ix

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