Explanation
A firm can be said to be a pure monopoly when a single firm dominates the market and it does not have any rival nor does its products have close substitutes. But, in a case, when there are other available substitutes, the monopolist would have to compete for dollars from consumers (to increase the number of units the monopolist would have to reduce price).

For instance, railways can be thought of as a pure monopoly, since there is no other firm producing the same service. Consumers willing to travel by railways have to pay charges set by the service provider. The service provider of railways can control the market price up to a certain extent. If they continue to charge higher prices even after the threshold people would opt for other available modes such as buses, airplanes, and so on.

This can be better explained using concepts of cross-price elasticity. Cross-price elasticity can be defined as the percentage change in quantity demanded of a commodity with respect to the percentage change in the price of other commodities. If a product has close substitutes available, then the monopolist faces a higher cross-price elasticity of demand. That is the monopolist has less control over the market (consumers would have alternatives to choose from). Again, when cross-price elasticity is less than one the monopolist has control over the price (consumers do not have many alternatives available to choose from). Consequently, it can be said that the monopolist faces competition if there are close substitutes available and they compete for the extra dollars from the consumers, which justifies the first statement but the second statement is only partially true.

Verified Answer
A monopolist does compete for extra dollars from consumers, but it does not imply that the non-existence of pure monopoly.

In case, a good has close substitutes available, that is, the cross-price elasticity is positive, it has to compete for extra dollars with the producers of substitute goods. On the other hand, if there are no such options or they are not convenient the producer can exercise its monopoly to a greater extent.