In topology, a closed map is a function between two topological spaces which maps closed sets to closed sets. That is, a function f : X → Y is closed if for any closed set U in X, the image f(U) is closed in Y. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.
Questions tagged [closed-map]
164 questions
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Function is Closed?
While reading the optimization textbook, the below proposition given:
Let $f: X \mapsto [-\infty, \infty]$ be a function. If $ \text{dom}(f)$ is closed and $f$ is lower semicontinuous at each x$\in \text{dom}(f)$, then $f$ is closed.
dom($f$) refers…
Beverlie
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T is a closed map if dim(Y/T(X)) is finite
I have to prove this statement:
Let $X$ and $Y$ be banach spaces and let $T \in L(X, Y)$ be a linear continuous operator. If $dim (Y/T(X)) < \infty $, then $T(X)$ is closed in Y.
My ideas:
$dim (Y/T(X)) = dim(Y) - dim(T(X))=: N$. But how can I…
MathStudent
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