Definition
A topological space $X$ is locally connected if each point $x\in X$ has a base of connected neighborhoods.
So let be $X$ a locally connected space and let be $Y$ a connected subspace. So if $\mathcal B(x)$ is a local connected base for any $x\in Y$ then the collection $\mathcal B'(x):=\big\{B_x\cap Y:B_x\in\mathcal B(x)\big\}$ is a local base with respect subspace topology on $Y$ and thus the statement follows proving that this base is locally connected but unfortunately I did not able to do this so that I think that the statement is false becouse after all the intersection between two connected set could be not connected however I did not find any counterexample showing that the local connectedness is not hereditable over the connected subspaces. So couldsome one help me, please?
In fact, every topological space is a subspace of a locally connected space. To see this, let $X$ be a topological space and let $Y$ be $X$ together with one new point $*$, topologized such that a subset of $Y$ is open iff it is either empty or has the form $U\cup\{*\}$ where $U$ is an open subset of $X$. Then any two nonempty open subsets of $Y$ intersect, so $Y$ is rather trivially locally connected, and the subspace topology on $X$ is the topology we started with.