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Download e-book for iPad: Curso de topología general by Francisco J. Díaz, José M. García Calcines

By Francisco J. Díaz, José M. García Calcines

ISBN-10: 8497708954

ISBN-13: 9788497708951

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Example text

En este caso, no es necesario que dicha familia sea finita. 6. 7 Sea (X, T ) un espacio topol´ ogico y {Ai }i∈I ⊂ T Ai . Si para todo i ∈ I ocurre que fi : (Ai , TAi ) → (X , T ) con X = i∈I CAP´ITULO 3. CONTINUIDAD Y CONVERGENCIA 52 es una aplicaci´ on continua y fi |Ai ∩Aj = fj |Ai ∩Aj para i = j, entonces f : (X, T ) → (X , T ) definida por f (x) = fi (x) si x ∈ Ai es tambi´en continua. 2. Aplicaciones abiertas y cerradas Adem´as de la continuidad las aplicaciones pueden verificar otras propiedades relacionadas con la topolog´ıa, que ser´an de utilidad, como veremos en cap´ıtulos posteriores.

1 se tiene que A ∈ Ent(x). Como x ∈ Cl(Y ) = X, de la definici´ on de clausura se deduce que que A ∩ Y = ∅. (⇐) Sea x ∈ X. Para todo U ∈ Ent(x), existe A ∈ T tal que x ∈ A ⊂ U . Por hip´ otesis A ∩ Y = ∅, de donde U ∩ Y = ∅. Luego todo punto de X pertenece a Cl(Y ), y por tanto Y es denso. 32 ´ CAP´ITULO 1. ESPACIOS TOPOLOGICOS (b) Consecuencia directa de que Cl(Y ) ∪ Ext(Y ) = X y Cl(Y ) ∩ Ext(Y ) = ∅. (c) (⇒) Si Y es denso y cerrado entonces Y = Cl(Y ) = X. (⇐) Si Y = X entonces Cl(Y ) = Y = X.

De lo hecho hasta ahora se desprende que no todo abierto en una topolog´ıa inducida es abierto en la topolog´ıa de partida. 9 Considerando la topolog´ıa cofinita sobre un conjunto infinito X y un subconjunto finito Y ⊂ X, es claro que Y ∈ (Tcof )Y pero Y no es abierto en (X, Tcof ). 10 En la topolog´ıa usual de la recta real se tiene que [0, 1] ∈ (Tu )[0,1] pero [0, 1] ∈ / Tu . Sin embargo, cuando el subconjunto Y es abierto en la topolog´ıa T de X, s´ı que ocurre que los abiertos de TY son exactamente los abiertos de T que est´an contenidos en Y .

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Curso de topología general by Francisco J. Díaz, José M. García Calcines

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