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In the uncountable, the least cardinal $k$ such that $X$ embeds into $I^k$ is simply the weight of the space $X$, i.e., the least size of a basis of the topology. A collection $\mathcal B$ is a basis for a topology $\tau$ if $\mathbb B\subseteq\tau$ and every et in $\tau$ is the union of members of $\mathcal B$.

In the light of mathahada's comment, it is interesting to note that this notion of dimension actually makes sense for zero-dimensional spaces. In the case of 0-dim spaces you might want to consider embedding in to $2^k$ rather than $I^k$, though. Now, zero-dimensional spaces are called zero-dimensional for a reason, but I don't think that giving $2^{k}$ dimension $k$ is necessarily pathological.

In any case, for spaces that don't embed into $I^{\aleph_0}$, i.e., for spaces of uncountable weight, you have defined something like a local weight that I haven't seen before. (Not that this would mean anything.)

Edit: Note that in the compact case, this local weight is just the weight: for each point choose an open neighborhood of minimal weight and cover the whole space by finitely many of these. Now weight of the whole space is the maximum of the weights of the finitely many oopne sets. (This argument assumes that we are dealing with infinite spaces.)

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In the uncountable, the least cardinal $k$ such that $X$ embeds into $I^k$ is simply the weight of the space $X$, i.e., the least size of a basis of the topology. A collection $\mathcal B$ is a basis for a topology $\tau$ if $\mathbb B\subseteq\tau$ and every et in $\tau$ is the union of members of $\mathcal B$.

In the light of mathahada's comment, it is interesting to note that this notion of dimension actually makes sense for zero-dimensional spaces. In the case of 0-dim spaces you might want to consider embedding in to $2^k$ rather than $I^k$, though. Now, zero-dimensional spaces are called zero-dimensional for a reason, but I don't think that giving $2^{k}$ dimension $k$ is necessarily pathological.

In any case, for spaces that don't embed into $I^{\aleph_0}$, i.e., for spaces of uncountable weight, you have defined something like a local weight that I haven't seen before. (Not that this would mean anything.)