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- mathematischer Satz (de)
- theorem (en)
- teorema matemàtic (ca)
- teorema (it)
- twierdzenie analizy funkcjonalnej o przestrzeniach Frécheta (pl)
- stelling uit de functionaalanalyse (nl)
- proposición topológica sobre funciones cerradas y continuas (es)
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- A cubic function (en)
- The Heaviside function (en)
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- The graph of the cubic function on the interval is closed because the function is continuous. The graph of the Heaviside function on is not closed, because the function is not continuous. (en)
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- Dirac distribution CDF.svg (en)
- cubicpoly.png (en)
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- If is a map from a topological space into a Hausdorff space then the graph of is closed if is continuous. The converse is true when is compact. (en)
- A linear map between two F-spaces is continuous if and only if its graph is closed. (en)
- For a Hausdorff compact range space , a set-valued function has a closed graph if and only if it is upper hemicontinuous and is a closed set for all . (en)
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- Theorem (en)
- Closed graph theorem (en)
- Closed graph theorem for set-valued functions (en)
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- First part: just note that the graph of is the same as the pre-image where is the diagonal in .
Second part:
For any open , we check is open. So take any , we construct some open neighborhood of , such that .
Since the graph of is closed, for every point on the "vertical line at x", with , draw an open rectangle disjoint from the graph of . These open rectangles, when projected to the y-axis, cover the y-axis except at , so add one more set .
Naively attempting to take would construct a set containing , but it is not guaranteed to be open, so we use compactness here.
Since is compact, we can take a finite open covering of as .
Now take . It is an open neighborhood of , since it is merely a finite intersection. We claim this is the open neighborhood of that we want.
Suppose not, then there is some unruly such that , then that would imply for some by open covering, but then , a contradiction since it is supposed to be disjoint from the graph of . (en)
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- Proof (en)
- Proof of closed graph theorem (en)
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- ProofOfClosedGraphTheorem (en)
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- Closed graph theorem (en)
- Teorema de la gràfica tancada (ca)
- Satz vom abgeschlossenen Graphen (de)
- Teorema de la gráfica cerrada (es)
- Théorème du graphe fermé (fr)
- Teorema del grafico chiuso (it)
- 닫힌 그래프 정리 (ko)
- 閉グラフ定理 (ja)
- Twierdzenie o wykresie domkniętym (pl)
- Stelling van de gesloten grafiek (nl)
- Teorema do gráfico fechado (pt)
- Satsen om den slutna grafen (sv)
- Теорема о замкнутом графике (ru)
- Теорема Банаха про замкнений графік (uk)
- 閉圖像定理 (zh)
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