It is well known that one of the major advantages of neural networks lies in their constant time response once they are trained. However, if a neural network includes a MAXNET [5], a neural net for selection of maximum values, it usually cannot have constant time response due to the drawback of inconsistent convergent time. Unfortunately, a neural network for classification often needs to use a MAXNET as a subnet. As a result, a MAXNET can be a bottleneck in the recalling phase, so that it is impractical to combine it with other neural networks. Most importantly, when a MAXNET serves as an intermediary subnet, the neural network may be led in a wrong decision in case that there are several largest values in the intermediate results, due to the random selection among those valules of the MAXNET. In this paper, a MAXNET with constant-time response and with multiple selection of several maximum values, CTMAXNET, is presented. The constant-time response for CTMAXNET is proved. The space and time complexities of CTMAXNET and related networks are presented and summarized. Some applications of CTMAXNETare presented to show the significance of CTMAXNET. From the example of ART1, it is shown that including MAXNET leads to misclassification while including CTMAXNET does not. From these results, a case is made that CTMAXNET is worthy of embedding in a large number of neural network classifier models and applications.