A space is indiscrete if the only open sets are the empty set and itself. As for the indiscrete topology, every set is compact because there is … More generally, any nite topological space is compact and any countable topological space is Lindel of. Prove that if Ais a subset of a topological space Xwith the indiscrete topology then Ais a compact subset. 6. Hence prove, by induction that a nite union of compact subsets of Xis compact. In derived categories in the homotopy-category sense (e.g. MATH31052 Topology Problems 6: Compactness 1. In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Removing just one element of the cover breaks the cover. 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. A space is compact … In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. 2. Prove that if K 1 and K 2 are compact subsets of a topological space X then so is K 1 [K 2. 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. discrete) is compact if and only if Xis nite, and Lindel of if and only if Xis countable. In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. A topological space (X;T) is called metrisable, if there exists a metric on Xsuch that the topology Tis induced by this metric. This is … 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. 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. $\begingroup$ @R.vanDobbendeBruyn In almost all cases I'm aware of, the abstract meaning coincides with the concrete meaning. So you can take the cover by those sets. 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.. Indiscrete or trivial. Compactness. Such a space is said to have the trivial topology. triangulated categories) directed colimits don't generally exist, so you're talking about a different notion anyway. The discrete topology on Xis metrisable and it is actually induced by the discrete metric. Compact. Every subset of X is sequentially compact. 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