I am currently trying to make an image for Topology. Although it doesn't really matter for the content, I think a smooth neighborhood looks nicer.
What I currently have

\documentclass[varwidth=true, border=2pt]{standalone}
\usepackage{tikz}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}[>=stealth,]
\begin{axis}[
axis x line=middle,
axis y line=middle,
%width=9cm,
%height=4.5cm,
xmin=-1, % start the diagram at this x-coordinate
xmax= 5, % end the diagram at this x-coordinate
ymin=-1, % start the diagram at this y-coordinate
ymax= 5, % end the diagram at this y-coordinate
xlabel=$X_1$,
ylabel=$X_2$,
ticks=none,
enlargelimits=true,
after end axis/.code={
\draw [decorate,decoration={brace,mirror,raise=15pt}] (axis cs:0,3.6) -- (axis cs:0,2.5) node [midway,left=20pt] {$U_2$};
\draw [decorate,decoration={brace,mirror,raise=12pt}] (axis cs:1.5,0) -- (axis cs:2.5,0) node [midway,below=16pt] {$U_1$};
}]
\addplot[mark=none, orange, smooth, thick, fill=orange!30] coordinates {(1,1) (2,0.5) (3,1.5) (3,2) (3.5,3) (3.2, 5) (2.2, 4.7) (1.5, 4.2) (1.1, 3.9) (0.9, 2.5) (1,1)};
\node[orange] at (axis cs:4,4) [anchor=south] {$U$};
% Draw help lines
\addplot[dashed] coordinates {(1.5,0) (1.5,3.6)};
\addplot[dashed] coordinates {(2.5,0) (2.5,3.6)};
\addplot[dashed] coordinates {(0,2.5) (2.5,2.5)};
\addplot[dashed] coordinates {(0,3.6) (2.5,3.6)};
% Draw solid square
\addplot[mark=none, red, thick, fill=red!30] coordinates {(2.5,2.5) (2.5,3.6) (1.5,3.6) (1.5,2.5) (2.5,2.5)};
% Draw x and annotation
\node[blue] at (axis cs:2,3) [anchor=south west] {$x$};
\addplot[mark=*, blue] coordinates {(2,3)};
% Draw ticks of help lines
\addplot[mark=none, red, thick] coordinates {(1.5, -0.1) (1.5,0.1)};
\addplot[mark=none, red, thick] coordinates {(2.5, -0.1) (2.5,0.1)};
\addplot[mark=none, red, thick] coordinates {(-0.1, 2.5) (0.1,2.5)};
\addplot[mark=none, red, thick] coordinates {(-0.1, 3.6) (0.1,3.6)};
% Draw axis text
\node[blue] at (axis cs:0,3) [anchor=east] {$x_2$};
\node[blue] at (axis cs:2,0) [anchor=north] {$x_1$};
\end{axis}
\end{tikzpicture}
\end{document}
What I want
I would like the orange neighborhood to be more smooth. The actual form of U doesn't matter, as long as the red square is included. How can I make U smoother?

smooth cycleinstead ofsmooth. If you usesmooth cycleyou have to remove the last coordinate (1,1) of your list. – Marco Daniel Oct 27 '13 at 08:41