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i would like to get help for these two questions.

1- Given $(X,\tau)$ be a topological space such that $\tau$ is finite. show that $X$ is compact.

2.consider $\mathbb{R}^2$ with the usual topology. show that $S_1=\lbrace\langle x,y\rangle|x^2+y^2=1\rbrace$ is compact subspace of $\mathbb R^2$. all the best

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2 Answers 2

Hint for 1: Let $\lbrace U_i\rbrace_{i\in I}$ be an open covering of $X$. What can you say about this covering? What property of $\tau$ tells you this covering is evidence of compactness?

Hint for 2: Since $\mathbb R^2$ in the usual topology is metrizable, it suffices to show that $S_1$ is sequence-compact. If $\lbrace \langle x_n,y_n\rangle\rbrace_{n\in\mathbb N}$ is a sequence in $S_1$ what can you say about $\lim_n\langle x_n,y_n\rangle$?

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For 2, if you know that $[0,1]$ is compact, you can write the circle as a continuous image of it, using a standard parametrisation. And images of compact spaces are compact.

Or show it is a closed subset of the compact $[-1,1] \times [-1,1]$, and use (if you know this already) that products of compact spaces are compact.

The last of course is already the idea of the proof of the Heine-Borel theorem, which also applies.

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