Why do indices show the previous day's closing price as the opeing price?  For example S&P500 Index.

This is not a true reflection of activity during the day.  It ignores the gaps that have been occuring in recent years.  On the chart, with an closing price say of 1378 for the S&P500 Index, that will also be the opening price the next day even though the index was never valued there.  The next day, the index value opened at 1371.

I realize that a true index calculation can not take place until all the stocks are opened but and some stocks do not always open or start trading on time due to imbalances or other reasons, but, the actual gap open for the index should be shown.

Many people do not actually realize that the index never traded that high, it becomes a misrepresentation of market activity because the high and open, or the low and open, can be mis-stated, and when we download your data, we back test ideas based on your data, we esentially are getting information that does not really represent the true market activity.

can you do anything to stop publishing the previous days close as the open.  Because this is not correct and truely is a misrepresentation of the actual market activity that occured.

The open of an index does not always reflect the closing price of the previous day. All you need to do is scroll back more than a single bar on your SP-500 example chart to see this.

The open of 4/23 and the close of 4/20 did in fact match at 1378.53, but the open of 4/20 was 1376.96 and the close of 4/19 was 1376.92. These values match other sources.