Let $ d, k in mathbb {N} $. A $ d $-dimensional $ C ^ k $-manifold is a topological space $ M $ equipped with a set $ (O ^ i, k ^ i) _ {i in mathcal {I}} $, with, for each $ i in mathcal {I} $, $ O ^ i $ is an open subset of $ M $ and $ k ^ i $ a map of $ O ^ i $ in $ mathbb {R} ^ {d} $ such as:
i) open sets $ O ^ i $, $ i in mathcal {I} $, more than $ M $;
ii) $ k ^ i (O ^ i) $ is an open subset of $ mathbb {R} ^ {d} $;
iii) for each $ i in mathcal {I} $ the map $ k ^ i: Oî longrightarrow k ^ i (O ^ i) $ is a homeomorphism;
iv) if $ O ^ {ij} = O ^ i cap O ^ j neq emptyset $Then for $ W ^ l = k ^ l (O ^ {ij}) subset k ^ l (O ^ l) $, $ l = i $ or $ j, the map $ k ^ {ij}: W ^ i longrightarrow W ^ j $ given by $ k ^ j circ (k ^ i) ^ {- 1} $ is a $ C ^ k $-difféomorphisme.
Definition: A variety $ M $ is $ sigma- $compact if there is a compact game sequence $ K ^ 0 subset K ^ 1 subset dots subset K ^ n subset dots $ such as $$ K ^ n subset operatorname {int} K ^ {n + 1} hbox {and} M = bigcup_ {n in mathbb {N}} K ^ n. $$
With this definition, can I say that all compact manifold $ M = K $ is $ sigma $-compact?
My attempt: taking $ K ^ n = K $ for everyone $ n in mathbb {N} $ we obtain $ operatorname {int} K ^ {n + 1} = operatorname {int} K = K $, since $ K = operatorname {int} K $ by the definition of the topology. Therefore, $ K ^ n = K subset K = operatorname {int} K ^ {n + 1} $. There seems to be something wrong.