Compact
Posted: Sun May 29, 2016 10:28 pm
Let \(\displaystyle{\left(X,\mathbb{T}\right)}\) be a topological space. If \(\displaystyle{\left(x_{n}\right)_{n\in\mathbb{N}}}\) is
a sequence of elements of \(\displaystyle{X}\) such that \(\displaystyle{x_{n}\longrightarrow x\,,n\longrightarrow +\infty}\), then prove
that the set \(\displaystyle{A=\left\{x_{n}:n\in\mathbb{N}\right\}\cup\left\{x\right\}\subseteq X}\) is a compact subset of \(\displaystyle{\left(X,\mathbb{T}\right)}\) .
a sequence of elements of \(\displaystyle{X}\) such that \(\displaystyle{x_{n}\longrightarrow x\,,n\longrightarrow +\infty}\), then prove
that the set \(\displaystyle{A=\left\{x_{n}:n\in\mathbb{N}\right\}\cup\left\{x\right\}\subseteq X}\) is a compact subset of \(\displaystyle{\left(X,\mathbb{T}\right)}\) .