I am trying to prove that local connectedness is a topological property. Then what do I need to show? Do I need to show that if $X$ and $Y$ are homeomorphic topological spaces (let $f$ be the homeomorphism) and $X$ is locally connected at $x$ in $X$ then $Y$ is locally connected at $f(x)$ or do I need to show that if $X$ is locally connected (i.e. $X$ is locally connected at each point in $X$) then $Y$ is also locally connected (i.e. $Y$ is locally connected at each point in $Y$)?
Also provide me some hints on how to do the proof.
Thanks in advance!
You have to show that if $X$ is a locally connected topological space and if $Y$ is a topological space homeomorphic to $X$, then $Y$ is locally connected too. In order to do that, let $f\colon X\longrightarrow Y$ be a homeomorphism. You are assuming that, for each $x\in X$, $x$ has a neighborhood basis $\mathcal N$ consisting entirely of open, connected sets. And you want to prove that the same thing holds in $Y$. So, take $y\in Y$, take $x=f^{-1}(y)$, take a neighborhood basis $\mathcal N$ of $x$ consisting entirely of open, connected sets, and prove that $\{f(V)\mid C\in\mathcal N\}$ is a neighborhood basis $\mathcal N$ of $y$ consisting entirely of open, connected sets.