# indiscrete topology is compact

Prove that if K 1 and K 2 are compact subsets of a topological space X then so is K 1 [K 2. $\begingroup$ @R.vanDobbendeBruyn In almost all cases I'm aware of, the abstract meaning coincides with the concrete meaning. 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. 2. In fact no infinite set in the discrete topology is compact. So you can take the cover by those sets. triangulated categories) directed colimits don't generally exist, so you're talking about a different notion anyway. A space is indiscrete if the only open sets are the empty set and itself. Such a space is said to have the trivial topology. Every subset of X is sequentially compact. On the other hand, the indiscrete topology on X is … 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. Compactness. A topological space (X;T) is called metrisable, if there exists a metric on Xsuch that the topology Tis induced by this metric. Hence prove, by induction that a nite union of compact subsets of Xis compact. A space is compact … 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. 6. 5.For any set X, (X;T indiscrete) is compact. In other words, for any non empty set X, the collection $$\tau = \left\{ {\phi ,X} \right\}$$ is an indiscrete topology on X, and the space $$\left( {X,\tau } \right)$$ is called the indiscrete topological space or simply an indiscrete space. In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Indiscrete or trivial. MATH31052 Topology Problems 6: Compactness 1. In the discrete topology, one point sets are open. The discrete topology is the finest topology that can be given on a set, i.e., it defines all subsets as open sets. Compact. In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. 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. discrete) is compact if and only if Xis nite, and Lindel of if and only if Xis countable. Removing just one element of the cover breaks the cover. [0;1] with its usual topology is compact. More generally, any nite topological space is compact and any countable topological space is Lindel of. As for the indiscrete topology, every set is compact because there is … This is … Every function to a space with the indiscrete topology is continuous . In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. In derived categories in the homotopy-category sense (e.g. 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.. Prove that if Ais a subset of a topological space Xwith the indiscrete topology then Ais a compact subset. The discrete topology on Xis metrisable and it is actually induced by the discrete metric. 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