Smooth Manifolds by Rajnikant Sinha

Author:Rajnikant Sinha
Language: eng
Format: epub
Publisher: Springer India, New Delhi

for some , where stands for: There exists a path in G from x to a. We have to show that [a] is an open subset of X, that is, every point of [a] is an interior point of [a]. For this purpose, let us take any . We must find an open neighborhood V of x such that .

Since and , . Since and G is open in X, G is an open neighborhood of x. Since X is a locally path connected space, and G is an open neighborhood of x, there exists an open neighborhood V of x such that(I)

(II) V is a path connected subset of X.

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