Relative compact inside a subset with relative topology
Clash Royale CLAN TAG #URR8PPP up vote 1 down vote favorite I am a little bit confused. Assume that $Y$ is a subset of $X$, with the inherited topology from $X$. What does it mean the $Ksubset Y$ is a relatively compact subset of $Y$? One option, is that $K= Ccap Y$, where $C$ is a relatively compact subset in $X$. Another option, is the $overlineKcap Y$ is compact in $Y$. To be specific, I am looking on the following example: Let $X$ be a topological (locally compact Hausdorff), $D_n_ ninmathbbZ$ be a sequence of open subsets inside $X$. For every $n$, let $T_n: D_nto D_-n$ be a homeomorphism. Consider the set $Y:= xin D_n$ with the inherited topology from $Xtimes mathbbZtimes X$. I want to say that if $Ksubset Y$ is relatively compact, then $K$ is contained in a set of the form $(T_n(x),n,x)in Xtimes mathbbZtimes X$ for some relatively compact set $C$ in $X$ and $NinmathbbN$. Is it true? Thanks! general-topology compactness share | cite | improve this question ...