Melania Trump's Theorem

In my previous blog entry, I used Melania Trump's theorem to prove a claim about continuous images in Hausdorff spaces of limit points and got ridiculed and accused of plagiarism. Therefore, I will formally state and prove her theorem in this blog entry as a way to apologize.

Melania Trump's Theorem: If $X$ and $Y$ are topological spaces, $A\subseteq X$, and $f:X\to Y$ is continuous, then $f(\overline A)\subseteq\overline{f\left(A\right)}$.

Proof: Let $y\in f(\overline A)=f\left(A'\cup A\right)=f\left(A'\right)\cup f\left(A\right)$ and suppose that $y\in f\left(A\right)$. Then $y\in\overline{f\left(A\right)}$. On the other hand, suppose that $y\in f\left(A'\right)$, let $V$ be a neighborhood of $y$, and let $x\in A'$ be such that $f\left(x\right)=y$. Then $f^{-1}\left(V\right)$ is a neighborhood of $x$, which implies that $f^{-1}\left(V\right)$ intersects $A$ at a point $x_0\neq x$, which in turn implies that $f\left(x_0\right)\in V$. Therefore, $y\in\overline{f\left(A\right)}$. $\blacksquare$

Trump University taught me better than to claim part of someone else's work as my own.

Note: This is mathematically sound satire.

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