As a result of this question, I've been thinking about the following condition on a topological space $Y$:
For every topological space $X$, $E\subseteq X$, and continuous maps $f,g\colon X\to Y$, if $E$ is dense in $X$, and $f$ and $g$ agree on $E$ (that is, $f(e)=g(e)$ for all $e\in E$), the...
This problem has a very straightforward solution when you conceptualize convergence of nets in terms of continuity of maps. Given a directed set $I$, the statement that a net $(y_i)_{i\in I}$ in a space $Y$ converges to a point $y$ can be expressed in terms of continuity of a map. Namely, let $...
This problem has a very straightforward solution when you conceptualize convergence of nets in terms of continuity of maps. Given a directed set $I$, the statement that a net $(y_i)_{i\in I}$ in a space $Y$ converges to a point $y$ can be expressed in terms of continuity of a map. Namely, let $...
