Removing just one element of the cover breaks the cover. The collection of the non empty set and the set X itself is always a topology on X, and is called the indiscrete topology on X. On the other hand, the indiscrete topology on X is … So you can take the cover by those sets. Compactness. 5.For any set X, (X;T indiscrete) is compact. More generally, any nite topological space is compact and any countable topological space is Lindel of. [0;1] with its usual topology is compact. Such spaces are commonly called indiscrete, anti-discrete, or codiscrete.Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Prove that if K 1 and K 2 are compact subsets of a topological space X then so is K 1 [K 2. 2. The discrete topology on Xis metrisable and it is actually induced by the discrete metric. In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. Indiscrete or trivial. A topological space (X;T) is called metrisable, if there exists a metric on Xsuch that the topology Tis induced by this metric. A space is compact … MATH31052 Topology Problems 6: Compactness 1. This is … $\begingroup$ @R.vanDobbendeBruyn In almost all cases I'm aware of, the abstract meaning coincides with the concrete meaning. In the discrete topology, one point sets are open. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology.Another name for general topology is point-set topology.. Hence prove, by induction that a nite union of compact subsets of Xis compact. discrete) is compact if and only if Xis nite, and Lindel of if and only if Xis countable. In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. Every subset of X is sequentially compact. As for the indiscrete topology, every set is compact because there is … triangulated categories) directed colimits don't generally exist, so you're talking about a different notion anyway. In fact no infinite set in the discrete topology is compact. Prove that if Ais a subset of a topological space Xwith the indiscrete topology then Ais a compact subset. Such a space is said to have the trivial topology. Every function to a space with the indiscrete topology is continuous . Compact. X is path connected and hence connected but is arc connected only if X is uncountable or if X has at most a single point. A space is indiscrete if the only open sets are the empty set and itself. 6. 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