If you need dots (to mark a point/coordinate) it's always better to use an actual circle drawn by TikZ. The glyphs \bullet or \textbullet don't necessarily have their center on the actual coordinate and you can't reference them as good.
A node of shape circle is used best. Setting the inner sep to zero allows the diameter to be given via minimum size.
To put a node closer to that dot it be best to reduce its inner sep. Technically, using a negative outer sep works, too, but I don't see a good reason for it in this case. (See the inner sep value in the every label key.)
Since it seems like you want to label a coordinate on the page, the label key is a nice way to place another node in reference to a given node without having to give it an explicit name. It allows you to use a style like dot in the second code which not only adds the label but also places the dot on the right coordinate without having to give it twice. (Though, this will not always work, of course.)
Code
\documentclass[tikz]{standalone} \begin{document} \tikz[ bullet/.style={shape=circle, fill, draw, inner sep=+0pt, minimum size=+3.5pt}, every label/.append style={inner sep=+.1em} ] \node[bullet, red, label={north east:$(1, 0)$}] at (1,0) {}; \tikz[ dot/.style args={#1:#2}{bullet, label={#1:$#2$}, at={#2}, node contents=}, bullet/.style={shape=circle, fill, draw, inner sep=+0pt, minimum size=+3.5pt}, every label/.append style={inner sep=+.1em} ] \node[red, dot={north east:(1, 0)}]; \end{document}
inner sepvalue of one of those nodes (or technically a negativeouter sepvalue would work also). It might be better to use afilledcirclenode instead of\bulletfor more control.